Home / Biology / The Living World
Cardinality of Sets
Cardinality of Sets, Singleton and Null Sets
Cardinality of Sets, Subset and Superset
Set Builder Form
Set Operations
Singleton and Null Sets
Subset and Superset
Union and Intersection of Sets
Union and Intersection of Sets, Complement of a Set, Subset and Superset
Union and Intersection of Sets, Subset and Superset, Venn Diagrams
Venn Diagrams
Understanding the Cardinality of a Set
The cardinality of a set is a measure of the "number of elements" in the set. For finite sets, the cardinality is simply the count of elements in the set. For infinite sets, cardinality tells us about the "size" of the infinity, distinguishing between different types of infinite sets (e.g., countably infinite vs. uncountably infinite sets).
Examples of Cardinality
1. Finite Set Example:
- Let \( A = \{1, 2, 3, 4\} \).
- The cardinality of \( A \), denoted \( |A| \), is the count of elements in \( A \).
- Solution: \( |A| = 4 \).
2. Empty Set Example:
- The empty set \( \emptyset \) has no elements.
- Solution: \( |\emptyset| = 0 \).
3. Union of Two Sets:
- Let \( B = \{1, 2, 3\} \) and \( C = \{3, 4, 5\} \).
- To find the cardinality of \( B \cup C \):
\(|B \cup C| = |B| + |C| - |B \cap C|\)
- Here, \( |B| = 3 \), \( |C| = 3 \), and \( |B \cap C| = 1 \) (since \( \{3\} \) is the only common element).
- Solution:
\( |B \cup C| = 3 + 3 - 1 = 5 \)
4. Cartesian Product Example:
- Let \( D = \{a, b\} \) and \( E = \{1, 2, 3\} \).
- The Cartesian product \( D \times E \) consists of all ordered pairs where the first element is from \( D \) and the second is from \( E \).
- Solution:
\[|D \times E| = |D| \cdot |E| = 2 \cdot 3 = 6\]
5. Power Set Example:
- Let \( F = \{x, y\} \).
- The power set of \( F \), denoted \( \mathcal{P}(F) \), is the set of all subsets of \( F \).
- Solution:
\(|\mathcal{P}(F)| = 2^{|F|} = 2^2 = 4\)
- Thus, \( \mathcal{P}(F) = \{\emptyset, \{x\}, \{y\}, \{x, y\}\} \), and \( |\mathcal{P}(F)| = 4 \).
Formulas to Find the Cardinality of Sets
Here's a list of formulas to determine the cardinality of various types of sets:
1. Cardinality of a Finite Set:
\(|A| = \text{number of elements in } A\)
2. Union of Two Sets \( A \) and \( B \):
\[ |A \cup B| = |A| + |B| - |A \cap B| \]
3. Union of Three Sets \( A \), \( B \), and \( C \):
\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C| \]
4. Complement of a Set \( A \):
\[ |A^c| = |U| - |A| \]
where \( U \) is the universal set containing \( A \).
5. Cartesian Product of Two Sets \( A \) and \( B \):
\[ |A \times B| = |A| \cdot |B| \]
6. Power Set of a Set \( A \):
\[ |\mathcal{P}(A)| = 2^{|A|} \]
7. Difference of Two Sets \( A \) and \( B \):
\[ |A - B| = |A| - |A \cap B| \]
8. Symmetric Difference of Two Sets \( A \) and \( B \):
\[ |A \triangle B| = |A| + |B| - 2|A \cap B| \]
9. Union of \( n \) Finite Sets \( A_1, A_2, \ldots, A_n \) (Generalized Inclusion-Exclusion Principle):
\[ \left|\bigcup_{i=1}^n A_i\right| = \sum_{k=1}^n (-1)^{k+1} \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq n} \left|A_{i_1} \cap A_{i_2} \cap \dots \cap A_{i_k}\right| \]
This formula adjusts for overlapping elements among multiple sets.
Notes on Infinite Sets
For infinite sets, cardinality is often discussed in terms of countable and uncountable infinities:
- Countably Infinite: Sets like \( \mathbb{N} \) (natural numbers) and \( \mathbb{Z} \) (integers) are countably infinite, meaning there exists a one-to-one correspondence with the natural numbers.
Want to master the topic? Log in to learne2i’s Student Hub for a personalized learning experience. Enhance your preparation with our IRT-based adaptive practice mode, which scales difficulty dynamically based on your performance to optimize learning outcomes.
Understanding Singleton and Null Sets
1. Singleton Set
A singleton set is a set that contains exactly one element. It has a cardinality of 1, meaning it contains only a single distinct element.
- Notation and Examples:
- A singleton set can be represented as \( \{a\} \), where \( a \) is the only element.
- Examples:
- \( S = \{5\} \): This set has only one element, 5.
- \( T = \{x \mid x = 0\} \): This set contains only one element, 0.
- \( U = \{\text{"apple"}\} \): This set has just the string "apple."
2. Null Set (Empty Set)
A null set or empty set is a set that contains no elements. Its cardinality is 0.
- Notation and Examples:
- The null set is denoted by \( \emptyset \) or \( \{\} \).
- Examples:
- \( V = \{\} \): This is an empty set with no elements.
- \( W = \{x \mid x < 0 \text{ and } x \in \mathbb{N}\} \): The set of natural numbers less than zero is an empty set, since no natural number is less than zero.
- \( X = \{x \mid x^2 = -1 \text{ and } x \in \mathbb{R}\} \): The set of real numbers whose square is -1 is empty, as no real number squared is negative.
Properties and Formulas Related to Singleton and Null Sets
Here are some properties and formulas for working with singleton and null sets:
1. Cardinality of a Singleton Set
- A singleton set \( \{a\} \) has a cardinality of 1:
\[
|\{a\}| = 1
\]
2. Cardinality of Null Set
- The null set \( \emptyset \) has a cardinality of 0:
\[
|\emptyset| = 0
\]
3. Union Involving Null and Singleton Sets
- The union of any set \( A \) with the null set is \( A \):
\[
A \cup \emptyset = A
\]
- For the union of a singleton set \( \{a\} \) with another set \( B \):
\[
|B \cup \{a\}| = \begin{cases}
|B| & \text{if } a \in B \\
|B| + 1 & \text{if } a \notin B
\end{cases}
\]
4. Intersection Involving Null and Singleton Sets
- The intersection of any set \( A \) with the null set is the null set:
\[
A \cap \emptyset = \emptyset
\]
- If \( A \) is a singleton set \( \{a\} \) and \( B \) is another set:
\[
A \cap B = \begin{cases}
\{a\} & \text{if } a \in B \\
\emptyset & \text{if } a \notin B
\end{cases}
\]
5. Power Set of Singleton and Null Sets
- The power set of a singleton set \( \{a\} \) is the set of all subsets of \( \{a\} \), which includes the null set and the singleton set itself:
\[
\mathcal{P}(\{a\}) = \{\emptyset, \{a\}\}
\]
with \( |\mathcal{P}(\{a\})| = 2 \).
- The power set of the null set \( \emptyset \) is the set containing only the null set:
\[
\mathcal{P}(\emptyset) = \{\emptyset\}
\]
with \( |\mathcal{P}(\emptyset)| = 1 \).
6. Subset Relationships
- The null set is a subset of every set:
\[
\emptyset \subseteq A
\]
- A singleton set \( \{a\} \) is a subset of a set \( A \) if \( a \in A \):
\[
\{a\} \subseteq A \iff a \in A
\]
Summary of Formulas
| Property | Formula/Relation |
|-----------------------------------------|----------------------------------------------|
| Cardinality of a Singleton Set \( \{a\} \) | \( |\{a\}| = 1 \) |
| Cardinality of Null Set \( \emptyset \) | \( |\emptyset| = 0 \) |
| Union with Null Set | \( A \cup \emptyset = A \) |
| Union of Singleton \( \{a\} \) and \( B \) | \( |B \cup \{a\}| = |B| \) if \( a \in B \); \( |B| + 1 \) if \( a \notin B \) |
| Intersection with Null Set | \( A \cap \emptyset = \emptyset \) |
| Intersection of Singleton \( \{a\} \) and \( B \) | \( A \cap B = \{a\} \) if \( a \in B \); \( \emptyset \) if \( a \notin B \) |
| Power Set of Singleton \( \{a\} \) | \( \mathcal{P}(\{a\}) = \{\emptyset, \{a\}\} \), \( |\mathcal{P}(\{a\})| = 2 \) |
| Power Set of Null Set \( \emptyset \) | \( \mathcal{P}(\emptyset) = \{\emptyset\} \), \( |\mathcal{P}(\emptyset)| = 1 \) |
| Null Set as a Subset | \( \emptyset \subseteq A \) |
| Singleton as a Subset | \( \{a\} \subseteq A \iff a \in A \) |
Cardinality of Sets
The cardinality of a set refers to the number of elements in the set. It provides a way to measure the "size" of both finite and certain types of infinite sets.
1. Cardinality of a Finite Set
- If \( A \) is a finite set, its cardinality \( |A| \) is simply the number of elements in \( A \).
- Example:
- Let \( A = \{2, 4, 6, 8\} \). Then, \( |A| = 4 \) because \( A \) has four elements.
2. Cardinality of Infinite Sets
- For infinite sets, cardinality describes the "type" of infinity:
- Countably Infinite: Sets that can be matched one-to-one with the natural numbers, like \( \mathbb{N} \) (natural numbers) or \( \mathbb{Z} \) (integers).
- Uncountably Infinite: Larger infinities, like the set of real numbers \( \mathbb{R} \), which cannot be matched one-to-one with the natural numbers.
3. Examples of Cardinalities
- Finite Set Example: \( B = \{a, b, c\} \). \( |B| = 3 \).
- Countably Infinite Set: \( \mathbb{N} = \{1, 2, 3, \ldots\} \). The cardinality of \( \mathbb{N} \) is denoted by \( \aleph_0 \) (aleph-null).
- Uncountably Infinite Set: The real numbers \( \mathbb{R} \) have a greater cardinality than \( \mathbb{N} \), often denoted as \( \mathfrak{c} \).
Subset
A subset is a set whose elements are all contained within another set. If every element of set \( A \) is also an element of set \( B \), then \( A \) is a subset of \( B \).
1. Notation
- We write \( A \subseteq B \) if \( A \) is a subset of \( B \).
- If \( A \subseteq B \) and \( A \neq B \), then \( A \) is called a proper subset of \( B \), written as \( A \subset B \).
2. Subset Properties
- Every set is a subset of itself: \( A \subseteq A \).
- The empty set \( \emptyset \) is a subset of every set: \( \emptyset \subseteq A \) for any set \( A \).
3. Example of Subsets
- Let \( C = \{1, 2, 3\} \) and \( D = \{1, 2, 3, 4, 5\} \).
- Here, \( C \subseteq D \) because every element of \( C \) is in \( D \).
- \( C \subset D \) since \( D \) has additional elements (4 and 5), making \( C \) a proper subset of \( D \).
Superset
A superset is the opposite of a subset. If every element of \( A \) is also in \( B \), then \( B \) is a superset of \( A \).
1. Notation
- We write \( B \supseteq A \) if \( B \) is a superset of \( A \).
- If \( B \supseteq A \) and \( B \neq A \), then \( B \) is called a proper superset of \( A \), written as \( B \supset A \).
2. Superset Properties
- Every set is a superset of itself: \( A \supseteq A \).
- Any set containing all elements of another set is its superset.
3. Example of Supersets
- Let \( E = \{1, 2, 3, 4, 5\} \) and \( F = \{1, 2, 3\} \).
- Here, \( E \supseteq F \) because \( E \) contains all elements of \( F \).
- \( E \supset F \) since \( E \) has additional elements (4 and 5), making \( E \) a proper superset of \( F \).
Certainly! I'll present the formulas and properties using a list format instead of a table.
Formulas and Properties Involving Cardinality, Subsets, and Supersets
1. Cardinality of a Finite Set
- Formula: \( |A| = \text{number of elements in } A \).
2. Union of Two Sets
- Formula: \( |A \cup B| = |A| + |B| - |A \cap B| \).
3. Union of Three Sets
- Formula:
\( |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C| \).
4. Subset Definition
- Condition: \( A \subseteq B \) if and only if every element of \( A \) is also an element of \( B \).
- Formula: \( A \subseteq B \iff \forall x (x \in A \Rightarrow x \in B) \).
5. Superset Definition
- Condition: \( B \supseteq A \) if and only if every element of \( A \) is also an element of \( B \).
- Formula: \( B \supseteq A \iff \forall x (x \in A \Rightarrow x \in B) \).
6. Number of Subsets of a Finite Set \( A \)
- If \( |A| = n \), then \( A \) has \( 2^n \) subsets.
7. Number of Proper Subsets of \( A \)
- The number of proper subsets (which excludes the set itself) is \( 2^n - 1 \).
8. Power Set of a Set
- The power set \( \mathcal{P}(A) \) of a set \( A \) is the set of all subsets of \( A \), including \( A \) itself and the empty set.
- Formula for Power Set Cardinality: \( |\mathcal{P}(A)| = 2^{|A|} \).
Additional Notes
- Subset and Superset Relations in a Universal Set \( U \):
- The null set \( \emptyset \) is a subset of every set within \( U \): \( \emptyset \subseteq A \).
- The universal set \( U \) is a superset of every set within it: \( U \supseteq A \).
Set Builder Form
The Set Builder Form is a method for describing a set by specifying a property that its members must satisfy. Rather than listing all elements, it defines the set in terms of a condition or rule.
General Format
- The set builder form for a set \( A \) can be written as:
\[
A = \{ x \mid \text{condition on } x \}
\]
- Here:
- \( x \): represents the elements of the set.
- \( \mid \): means "such that."
- Condition on \( x \): describes the property or condition that \( x \) must satisfy to be in the set.
Examples of Set Builder Form
1. Simple Example:
- Set of all even natural numbers:
\[
A = \{ x \mid x \text{ is an even natural number} \} = \{2, 4, 6, 8, \dots\}
\]
2. Specific Condition:
- Set of all integers greater than 0 and less than 10:
\[
B = \{ x \mid x \in \mathbb{Z}, 0 < x < 10 \} = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}
\]
3. Using Algebraic Conditions:
- Set of squares of natural numbers less than or equal to 25:
\[
C = \{ x \mid x = n^2, n \in \mathbb{N}, n \leq 5 \} = \{1, 4, 9, 16, 25\}
\]
4. Multiple Conditions:
- Set of all real numbers between -2 and 5, excluding -2 and 5:
\[
D = \{ x \mid x \in \mathbb{R}, -2 < x < 5 \}
\]
5. Infinite Sets Using Set Builder Form:
- Set of all natural numbers greater than 10:
\[
E = \{ x \mid x \in \mathbb{N}, x > 10 \} = \{11, 12, 13, \dots\}
\]
Symbols Commonly Used in Set Builder Form
1. \( \mid \) or \( : \)
- Meaning "such that." It separates the variable from the condition in set builder notation.
- Example: \( F = \{ x \mid x > 0 \} \).
2. \( \in \)
- Denotes "is an element of." Indicates the domain or type of elements.
- Example: \( G = \{ x \in \mathbb{R} \mid x^2 = 4 \} \).
3. Logical Connectives:
- \( \wedge \) (and): Used for combining multiple conditions.
- Example: \( H = \{ x \in \mathbb{Z} \mid x > 0 \wedge x < 10 \} \).
- \( \vee \) (or): Indicates that satisfying any one condition is sufficient.
- Example: \( I = \{ x \in \mathbb{Z} \mid x < -3 \vee x > 3 \} \).
4. Inequality Symbols:
- \( <, >, \leq, \geq \) to specify numerical ranges.
- Example: \( J = \{ x \in \mathbb{R} \mid -1 \leq x < 5 \} \).
Advantages of Set Builder Form
1. Concise Representation:
- Especially useful for representing large or infinite sets concisely, without listing all elements.
2. Flexibility with Conditions:
- Allows defining sets with complex conditions, like prime numbers, even numbers, or intervals.
3. Easier for Infinite Sets:
- Set builder notation makes it possible to describe infinite sets, such as all integers or real numbers within a range.
4. Clear Mathematical Descriptions:
- The form clarifies exactly what properties or conditions are required for elements in the set.
Using Set Builder Form with Different Types of Sets
1. Natural Numbers:
- Set of natural numbers greater than 5:
\[
K = \{ x \mid x \in \mathbb{N}, x > 5 \} = \{6, 7, 8, \dots\}
\]
2. Integer Sets:
- Set of odd integers:
\[
L = \{ x \mid x \in \mathbb{Z}, x \text{ is odd} \} = \{\dots, -3, -1, 1, 3, \dots\}
\]
3. Rational Numbers:
- Set of rational numbers between 0 and 1:
\[
M = \{ x \mid x \in \mathbb{Q}, 0 < x < 1 \}
\]
4. Real Number Intervals:
- Set of real numbers between -2 and 2, inclusive of endpoints:
\[
N = \{ x \mid x \in \mathbb{R}, -2 \leq x \leq 2 \}
\]
Summary of Set Builder Form Concepts
- General Notation: \( \{ x \mid \text{condition on } x \} \)
- Key Symbols:
- \( \mid \) or \( : \): "such that"
- \( \in \): "is an element of"
- \( \wedge \): "and"
- \( \vee \): "or"
- Advantages: Concise representation, handles infinite sets, accommodates complex conditions.
Set Operations
Set operations allow us to create new sets by combining, comparing, or altering existing sets. The primary set operations include Union, Intersection, Difference, and Complement.
1. Union of Sets
The union of two sets \( A \) and \( B \) is a set containing all elements that are in \( A \), in \( B \), or in both. The union operation combines elements from both sets without duplicating any elements.
- Notation: \( A \cup B \)
- Definition:
\[
A \cup B = \{ x \mid x \in A \text{ or } x \in B \}
\]
- Example:
Let \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \).
Then, \( A \cup B = \{1, 2, 3, 4, 5\} \).
2. Intersection of Sets
The intersection of two sets \( A \) and \( B \) is a set containing all elements that are both in \( A \) and in \( B \).
- Notation: \( A \cap B \)
- Definition:
\[
A \cap B = \{ x \mid x \in A \text{ and } x \in B \}
\]
- Example:
Let \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \).
Then, \( A \cap B = \{3\} \).
3. Difference of Sets
The difference of two sets \( A \) and \( B \) (also called the relative complement) is a set containing all elements that are in \( A \) but not in \( B \).
- Notation: \( A - B \) or \( A \setminus B \)
- Definition:
\[
A - B = \{ x \mid x \in A \text{ and } x \notin B \}
\]
- Example:
Let \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \).
Then, \( A - B = \{1, 2\} \) and \( B - A = \{4, 5\} \).
4. Complement of a Set
The complement of a set \( A \) is the set of all elements in the universal set \( U \) that are not in \( A \).
- Notation: \( A^c \) or \( \overline{A} \)
- Definition:
\[
A^c = \{ x \mid x \in U \text{ and } x \notin A \}
\]
- Example:
If the universal set \( U = \{1, 2, 3, 4, 5\} \) and \( A = \{1, 2, 3\} \),
then \( A^c = \{4, 5\} \).
5. Symmetric Difference
The symmetric difference of two sets \( A \) and \( B \) is a set containing all elements that are in either \( A \) or \( B \) but not in both.
- Notation: \( A \triangle B \)
- Definition:
\[
A \triangle B = (A - B) \cup (B - A)
\]
- Example:
Let \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \).
Then, \( A \triangle B = \{1, 2, 4, 5\} \).
Properties of Set Operations
1. Commutative Properties
- Union: \( A \cup B = B \cup A \)
- Intersection: \( A \cap B = B \cap A \)
2. Associative Properties
- Union: \( (A \cup B) \cup C = A \cup (B \cup C) \)
- Intersection: \( (A \cap B) \cap C = A \cap (B \cap C) \)
3. Distributive Properties
- \( A \cup (B \cap C) = (A \cup B) \cap (A \cup C) \)
- \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
4. Identity Properties
- Union with the empty set: \( A \cup \emptyset = A \)
- Intersection with the empty set: \( A \cap \emptyset = \emptyset \)
- Union with the universal set: \( A \cup U = U \)
- Intersection with the universal set: \( A \cap U = A \)
5. Complement Properties
- Double Complement: \( (A^c)^c = A \)
- De Morgan’s Laws:
- \( (A \cup B)^c = A^c \cap B^c \)
- \( (A \cap B)^c = A^c \cup B^c \)
Examples Using Set Properties
1. Using Commutative Property:
- If \( A = \{1, 2\} \) and \( B = \{2, 3\} \), then:
\[
A \cup B = B \cup A = \{1, 2, 3\}
\]
2. Applying Distributive Property:
- If \( A = \{1, 2\} \), \( B = \{2, 3\} \), and \( C = \{3, 4\} \), then:
\[
A \cup (B \cap C) = A \cup \{3\} = \{1, 2, 3\}
\]
\[
(A \cup B) \cap (A \cup C) = \{1, 2, 3\} \cap \{1, 2, 3, 4\} = \{1, 2, 3\}
\]
3. Using De Morgan's Laws:
- Let \( U = \{1, 2, 3, 4, 5\} \), \( A = \{1, 2\} \), and \( B = \{2, 3\} \). Then:
\[
(A \cup B)^c = A^c \cap B^c = \{3, 4, 5\} \cap \{1, 4, 5\} = \{4, 5\}
\]
Summary of Set Operations
- Union: Combines all elements from two sets without duplication.
- Intersection: Finds elements common to both sets.
- Difference: Finds elements in one set but not the other.
- Complement: Finds elements not in a set within a universal set.
- Symmetric Difference: Finds elements in either set but not in both.
Singleton Set
A singleton set is a set that contains exactly one element. This type of set has a cardinality of 1, meaning it includes only a single distinct element.
Characteristics of a Singleton Set
- A singleton set has only one element.
- Its cardinality (number of elements) is 1.
- Singleton sets are finite.
Notation and Examples
- A singleton set can be represented as \( S = \{a\} \), where \( a \) is the only element in the set.
- Examples:
- \( S = \{5\} \): This set contains only the element 5.
- \( T = \{x \mid x = 0\} \): This set has only one element, 0.
- \( U = \{\text{"apple"}\} \): This set contains the single string "apple."
Properties of Singleton Sets
1. Subset Property: Any singleton set \( \{a\} \) is a subset of any set that contains \( a \):
\[
\{a\} \subseteq A \iff a \in A
\]
2. Power Set: The power set of a singleton set \( \{a\} \), which is the set of all subsets of \( \{a\} \), includes the empty set and the singleton set itself:
\[
\mathcal{P}(\{a\}) = \{\emptyset, \{a\}\}
\]
The cardinality of \( \mathcal{P}(\{a\}) \) is 2.
---
Null Set (Empty Set)
A null set (or empty set) is a set that contains no elements. It is the unique set with a cardinality of 0, meaning it has no members.
Characteristics of the Null Set
- The null set is empty, with no elements.
- Its cardinality is 0.
- The null set is considered a subset of every set.
Notation and Examples
- The null set is commonly denoted by \( \emptyset \) or \( \{\} \).
- Examples:
- \( V = \{\} \): An empty set with no elements.
- The set of natural numbers less than zero: \( W = \{x \mid x < 0, x \in \mathbb{N}\} = \emptyset \), since no natural number is less than zero.
- The set of square roots of -1 in the real number system: \( X = \{x \mid x^2 = -1, x \in \mathbb{R}\} = \emptyset \), since no real number squared is negative.
Properties of the Null Set
1. Subset Property: The null set is a subset of every set:
\[
\emptyset \subseteq A
\]
2. Union with Any Set: The union of any set \( A \) with the null set is \( A \):
\[
A \cup \emptyset = A
\]
3. Intersection with Any Set: The intersection of any set \( A \) with the null set is the null set:
\[
A \cap \emptyset = \emptyset
\]
4. Power Set of the Null Set: The power set of the null set \( \emptyset \), which is the set of all subsets of \( \emptyset \), contains only the null set itself:
\[
\mathcal{P}(\emptyset) = \{\emptyset\}
\]
The cardinality of \( \mathcal{P}(\emptyset) \) is 1.
---
Comparing Singleton and Null Sets
- Cardinality:
- Singleton Set: Has a cardinality of 1.
- Null Set: Has a cardinality of 0.
- Number of Elements:
- Singleton Set: Contains exactly one element.
- Null Set: Contains no elements.
- Subset Property:
- Singleton Set: \( \{a\} \subseteq A \) if \( a \in A \).
- Null Set: \( \emptyset \subseteq A \) for any set \( A \).
- Union with Another Set:
- Singleton Set: The union \( A \cup \{a\} \) adds one element if \( a \notin A \).
- Null Set: The union \( A \cup \emptyset = A \), as the null set adds no elements.
- Intersection with Another Set:
- Singleton Set: The intersection \( A \cap \{a\} = \{a\} \) if \( a \in A \); otherwise, it is \( \emptyset \).
- Null Set: The intersection \( A \cap \emptyset = \emptyset \), as there are no common elements.
---
Examples and Use Cases
1. Singleton Set Example:
- Let \( C = \{7\} \).
- The power set of \( C \) is \( \mathcal{P}(C) = \{\emptyset, \{7\}\} \).
- \( C \) has only one element, so \( |C| = 1 \).
2. Null Set Example:
- Let \( D = \{x \mid x > 10, x < 5\} \).
- There is no number that simultaneously satisfies \( x > 10 \) and \( x < 5 \), so \( D = \emptyset \).
- The power set of \( D \) is \( \mathcal{P}(\emptyset) = \{\emptyset\} \), with \( |\mathcal{P}(\emptyset)| = 1 \).
3. Union and Intersection with Null Set:
- If \( E = \{2, 4, 6\} \), then:
- \( E \cup \emptyset = E = \{2, 4, 6\} \)
- \( E \cap \emptyset = \emptyset \)
4. Subset Properties:
- \( \emptyset \subseteq \{1, 2, 3\} \) (The null set is a subset of every set).
- \( \{5\} \subseteq \{3, 5, 7\} \) because 5 is an element of \( \{3, 5, 7\} \).
---
Summary of Singleton and Null Sets
- Singleton Set: A set with exactly one element, denoted by \( \{a\} \), where \( a \) is the only element.
- Properties: Has a cardinality of 1, is a subset of any set containing \( a \).
- Null Set (Empty Set): A set with no elements, denoted by \( \emptyset \).
- Properties: Has a cardinality of 0, is a subset of every set, and does not add elements when forming unions.
Subset
A subset is a set where every element of one set is also contained within another set. In other words, if all elements of set \( A \) are also elements of set \( B \), then \( A \) is a subset of \( B \).
Notation
- \( A \subseteq B \) denotes that \( A \) is a subset of \( B \).
- If \( A \) is a subset of \( B \) and \( A \neq B \), then \( A \) is called a proper subset of \( B \), written as \( A \subset B \).
Properties of Subsets
1. Reflexive Property:
- Every set is a subset of itself:
\[
A \subseteq A
\]
2. Empty Set as a Subset:
- The empty set \( \emptyset \) is a subset of every set:
\[
\emptyset \subseteq A
\]
3. Subset of a Universal Set:
- Every set is a subset of the universal set \( U \), which contains all elements in a given context:
\[
A \subseteq U
\]
4. Number of Subsets:
- If a set \( A \) has \( n \) elements, then the number of subsets of \( A \) is \( 2^n \), including the empty set and \( A \) itself.
5. Number of Proper Subsets:
- A proper subset excludes the set itself. So, if \( A \) has \( n \) elements, the number of proper subsets of \( A \) is \( 2^n - 1 \).
Examples of Subsets
1. Let \( A = \{1, 2, 3\} \) and \( B = \{1, 2, 3, 4, 5\} \).
- Here, \( A \subseteq B \) because every element of \( A \) is also in \( B \).
- Since \( A \neq B \), \( A \) is a proper subset of \( B \): \( A \subset B \).
2. Let \( C = \{a, b\} \).
- The subsets of \( C \) are: \( \emptyset \), \( \{a\} \), \( \{b\} \), and \( \{a, b\} \).
- Since \( C \) has 2 elements, it has \( 2^2 = 4 \) subsets, including itself and the empty set.
---
Superset
A superset is the opposite of a subset. If all elements of \( A \) are also elements of \( B \), then \( B \) is considered a superset of \( A \).
Notation
- \( B \supseteq A \) denotes that \( B \) is a superset of \( A \).
- If \( B \supseteq A \) and \( B \neq A \), then \( B \) is called a proper superset of \( A \), written as \( B \supset A \).
Properties of Supersets
1. Reflexive Property:
- Every set is a superset of itself:
\[
A \supseteq A
\]
2. Universal Set as a Superset:
- The universal set \( U \) is a superset of every set within the context of that universe:
\[
U \supseteq A
\]
3. Empty Set:
- Every set is a superset of the empty set:
\[
A \supseteq \emptyset
\]
Examples of Supersets
1. Let \( D = \{1, 2, 3, 4, 5\} \) and \( E = \{1, 2, 3\} \).
- Here, \( D \supseteq E \) because every element of \( E \) is in \( D \).
- Since \( D \neq E \), \( D \) is a proper superset of \( E \): \( D \supset E \).
2. Let \( F = \{a, b, c\} \) and \( G = \{\} \).
- \( F \supseteq G \) since every element of \( G \) (there are none) is also in \( F \).
- Every set is a superset of the empty set.
---
Subset and Superset Relations
1. Subset and Superset of the Same Set:
- For any set \( A \), \( A \subseteq A \) and \( A \supseteq A \).
2. Subset and Superset in Terms of Universal Set:
- In the context of a universal set \( U \), for any set \( A \) within \( U \):
- \( A \subseteq U \)
- \( U \supseteq A \)
3. Empty Set Relations:
- The empty set \( \emptyset \) is a subset of every set, and every set is a superset of \( \emptyset \):
\[
\emptyset \subseteq A \quad \text{and} \quad A \supseteq \emptyset
\]
4. Proper Subset and Superset:
- A proper subset \( A \subset B \) means \( A \) is a subset of \( B \), but \( A \neq B \).
- Similarly, a proper superset \( B \supset A \) means \( B \) is a superset of \( A \), but \( B \neq A \).
---
Examples of Counting Subsets and Supersets
1. Counting Subsets:
- If \( H = \{1, 2\} \), then \( H \) has \( 2^2 = 4 \) subsets: \( \{\}, \{1\}, \{2\}, \{1, 2\} \).
2. Counting Proper Subsets:
- The proper subsets of \( H = \{1, 2\} \) exclude \( H \) itself, so there are \( 2^2 - 1 = 3 \) proper subsets: \( \{\}, \{1\}, \{2\} \).
3. Supersets in the Context of Universal Set:
- Suppose the universal set \( U = \{1, 2, 3\} \) and \( I = \{1, 2\} \).
- \( U \supseteq I \) because every element of \( I \) is in \( U \).
- \( I \subseteq U \) as \( I \) is a subset of \( U \).
---
Summary of Subset and Superset Concepts
- Subset:
- A set \( A \) is a subset of \( B \) if every element of \( A \) is in \( B \).
- Notation: \( A \subseteq B \) (or \( A \subset B \) if \( A \) is a proper subset).
- Key Properties: Reflexivity (\( A \subseteq A \)), Empty Set as a subset of every set, Number of subsets of a set with \( n \) elements is \( 2^n \).
- Superset:
- A set \( B \) is a superset of \( A \) if every element of \( A \) is in \( B \).
- Notation: \( B \supseteq A \) (or \( B \supset A \) if \( B \) is a proper superset).
- Key Properties: Reflexivity (\( A \supseteq A \)), Universal set as a superset of every set, Every set is a superset of the empty set.
Union of Sets
The union of two or more sets combines all elements from each set, without duplicating any elements. The union operation gathers all elements present in any of the sets involved.
Notation
- The union of sets \( A \) and \( B \) is denoted as \( A \cup B \).
- For more than two sets, such as \( A \), \( B \), and \( C \), the union is written as \( A \cup B \cup C \).
Definition
- The union of two sets \( A \) and \( B \) is the set of all elements that are in \( A \), in \( B \), or in both:
\[
A \cup B = \{ x \mid x \in A \text{ or } x \in B \}
\]
Properties of Union
1. Commutative Property:
- \( A \cup B = B \cup A \)
2. Associative Property:
- \( (A \cup B) \cup C = A \cup (B \cup C) \)
3. Identity Property:
- \( A \cup \emptyset = A \)
- \( A \cup U = U \) (where \( U \) is the universal set)
4. Idempotent Property:
- \( A \cup A = A \)
Examples of Union
- Let \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \).
- The union \( A \cup B = \{1, 2, 3, 4, 5\} \), which combines all elements from both sets.
- If \( C = \{a, b\} \) and \( D = \{b, c\} \), then:
- \( C \cup D = \{a, b, c\} \)
---
Intersection of Sets
The intersection of two or more sets contains only the elements that are common to all sets involved.
Notation
- The intersection of sets \( A \) and \( B \) is denoted as \( A \cap B \).
- For more than two sets, such as \( A \), \( B \), and \( C \), the intersection is written as \( A \cap B \cap C \).
Definition
- The intersection of two sets \( A \) and \( B \) is the set of all elements that are in both \( A \) and \( B \):
\[
A \cap B = \{ x \mid x \in A \text{ and } x \in B \}
\]
Properties of Intersection
1. Commutative Property:
- \( A \cap B = B \cap A \)
2. Associative Property:
- \( (A \cap B) \cap C = A \cap (B \cap C) \)
3. Identity Property:
- \( A \cap U = A \) (where \( U \) is the universal set)
- \( A \cap \emptyset = \emptyset \)
4. Idempotent Property:
- \( A \cap A = A \)
Examples of Intersection
- Let \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \).
- The intersection \( A \cap B = \{3\} \), which is the only element common to both sets.
- If \( C = \{a, b, c\} \) and \( D = \{b, c, d\} \), then:
- \( C \cap D = \{b, c\} \)
---
Formulas for Union and Intersection
1. Union of Two Sets
- Formula:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
- This formula calculates the cardinality (number of elements) of the union of two sets by adding their individual cardinalities and subtracting the cardinality of their intersection to avoid double-counting.
2. Union of Three Sets
- Formula:
\[
|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C|
\]
- This formula adjusts for overlapping elements in three sets by adding the cardinalities of individual sets, subtracting the intersections of each pair, and adding back the intersection of all three sets.
3. General Formula for Union of \( n \) Sets (Inclusion-Exclusion Principle)
For the union of multiple sets, we use the Inclusion-Exclusion Principle to avoid over-counting elements that are in the intersections of more than one set.
- Formula:
\[
\left| \bigcup_{i=1}^n A_i \right| = \sum_{k=1}^n (-1)^{k+1} \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq n} \left| A_{i_1} \cap A_{i_2} \cap \dots \cap A_{i_k} \right|
\]
- This formula iterates through all possible intersections of sets, alternating between adding and subtracting cardinalities based on the number of sets in each intersection.
For practical purposes, this formula is commonly applied to unions of up to three sets.
---
Examples Using Formulas for Union and Intersection
1. Union of Two Sets:
- Let \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \).
- \( |A| = 3 \), \( |B| = 3 \), and \( |A \cap B| = 1 \) (since \( A \cap B = \{3\} \)).
- Using the formula:
\[
|A \cup B| = |A| + |B| - |A \cap B| = 3 + 3 - 1 = 5
\]
2. Union of Three Sets:
- Let \( A = \{1, 2\} \), \( B = \{2, 3\} \), and \( C = \{1, 3, 4\} \).
- \( |A| = 2 \), \( |B| = 2 \), \( |C| = 3 \).
- \( |A \cap B| = 1 \) (common element 2), \( |B \cap C| = 1 \) (common element 3), \( |C \cap A| = 1 \) (common element 1).
- \( |A \cap B \cap C| = 0 \) (no element common to all three).
- Using the formula:
\[
|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C|
\]
\[
|A \cup B \cup C| = 2 + 2 + 3 - 1 - 1 - 1 + 0 = 4
\]
---
Summary of Union and Intersection of Sets
- Union:
- Combines all elements from multiple sets without duplicating any elements.
- Notation: \( A \cup B \) for two sets.
- Properties: Commutative, associative, identity with the empty set.
- Formula (for two sets): \( |A \cup B| = |A| + |B| - |A \cap B| \).
- Intersection:
- Includes only elements common to all sets.
- Notation: \( A \cap B \) for two sets.
- Properties: Commutative, associative, identity with the universal set, idempotent.
- Formula: The size of intersections is typically used in formulas for union, particularly with the Inclusion-Exclusion Principle.
These operations and formulas allow for comprehensive analysis of combined sets and are fundamental in set theory and applications like probability and data science
Like union of sets do we have a general formula for intersection of sets?
There isn't a single "general formula" for the intersection of multiple sets in the same way there is for the union of sets because the intersection does not involve an additive inclusion-exclusion adjustment like the union. However, we can think about the intersection of sets in a systematic way for any number of sets.
Intersection of \( n \) Sets
For any collection of \( n \) sets \( A_1, A_2, \dots, A_n \), the intersection of all sets contains only the elements that are present in every one of those sets.
General Intersection Notation
- For \( n \) sets \( A_1, A_2, \dots, A_n \), the intersection is written as:
\[
\bigcap_{i=1}^n A_i = A_1 \cap A_2 \cap \dots \cap A_n
\]
- This set includes only those elements that are common to each of the \( n \) sets.
General Property of Intersection
- If \( B \subseteq A_i \) for each \( i = 1, 2, \dots, n \), then \( B \subseteq \bigcap_{i=1}^n A_i \).
Example with Three Sets
For three sets \( A \), \( B \), and \( C \), the intersection can be expressed as:
\[
A \cap B \cap C = \{ x \mid x \in A \text{ and } x \in B \text{ and } x \in C \}
\]
This set includes only elements that all three sets \( A \), \( B \), and \( C \) share.
Cardinality of Intersection
The cardinality of the intersection depends on the overlap of elements among the sets, which varies with the specific sets and their elements. There is no inclusion-exclusion formula for intersection as there is for union, because we are only interested in elements common to all sets, without needing to adjust for partial overlaps.
Special Cases
1. Intersection of Two Sets:
- The cardinality is simply \( |A \cap B| \), which can be counted directly if the elements of \( A \) and \( B \) are known.
2. Disjoint Sets:
- If the sets are disjoint (no elements in common), then \( A_1 \cap A_2 \cap \dots \cap A_n = \emptyset \).
3. Subset Relation:
- If \( A \subseteq B \), then \( A \cap B = A \).
Conclusion
The intersection of multiple sets doesn’t require a formula as complex as the union because it’s defined solely by elements common to all sets involved.
Union of Sets
The union of two or more sets combines all elements from each set without duplicating any elements. It gathers all elements that are present in any of the sets involved.
Notation
- The union of sets \( A \) and \( B \) is denoted as \( A \cup B \).
- For more than two sets, such as \( A \), \( B \), and \( C \), the union is written as \( A \cup B \cup C \).
Definition
- The union of two sets \( A \) and \( B \) is the set of all elements that are in \( A \), in \( B \), or in both:
\[
A \cup B = \{ x \mid x \in A \text{ or } x \in B \}
\]
Properties of Union
1. Commutative Property:
- \( A \cup B = B \cup A \)
2. Associative Property:
- \( (A \cup B) \cup C = A \cup (B \cup C) \)
3. Identity Property:
- \( A \cup \emptyset = A \)
- \( A \cup U = U \), where \( U \) is the universal set
4. Idempotent Property:
- \( A \cup A = A \)
Formula for Union
- Union of Two Sets:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
- Union of Three Sets:
\[
|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C|
\]
- General Formula for Union of \( n \) Sets (Inclusion-Exclusion Principle):
\[
\left| \bigcup_{i=1}^n A_i \right| = \sum_{k=1}^n (-1)^{k+1} \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq n} \left| A_{i_1} \cap A_{i_2} \cap \dots \cap A_{i_k} \right|
\]
---
Intersection of Sets
The intersection of two or more sets contains only the elements that are common to all the sets involved.
Notation
- The intersection of sets \( A \) and \( B \) is denoted as \( A \cap B \).
- For more than two sets, such as \( A \), \( B \), and \( C \), the intersection is written as \( A \cap B \cap C \).
Definition
- The intersection of two sets \( A \) and \( B \) is the set of all elements that are in both \( A \) and \( B \):
\[
A \cap B = \{ x \mid x \in A \text{ and } x \in B \}
\]
Properties of Intersection
1. Commutative Property:
- \( A \cap B = B \cap A \)
2. Associative Property:
- \( (A \cap B) \cap C = A \cap (B \cap C) \)
3. Identity Property:
- \( A \cap U = A \), where \( U \) is the universal set
- \( A \cap \emptyset = \emptyset \)
4. Idempotent Property:
- \( A \cap A = A \)
---
Complement of a Set
The complement of a set \( A \) contains all elements that are in the universal set \( U \) but not in \( A \). It represents the "opposite" of \( A \) within \( U \).
Notation
- The complement of \( A \) is denoted as \( A^c \) or \( \overline{A} \).
Definition
- The complement of \( A \) in the universal set \( U \) is defined as:
\[
A^c = \{ x \mid x \in U \text{ and } x \notin A \}
\]
Properties of Complement
1. Complement of the Universal Set:
- \( U^c = \emptyset \)
2. Complement of the Empty Set:
- \( \emptyset^c = U \)
3. Double Complement (Involution Property):
- \( (A^c)^c = A \)
4. De Morgan’s Laws:
- \( (A \cup B)^c = A^c \cap B^c \)
- \( (A \cap B)^c = A^c \cup B^c \)
---
Subset
A subset is a set where every element of one set is also contained within another set. If all elements of set \( A \) are also elements of set \( B \), then \( A \) is a subset of \( B \).
Notation
- \( A \subseteq B \) denotes that \( A \) is a subset of \( B \).
- If \( A \) is a subset of \( B \) and \( A \neq B \), then \( A \) is called a proper subset of \( B \), written as \( A \subset B \).
Properties of Subsets
1. Reflexive Property:
- Every set is a subset of itself:
\[
A \subseteq A
\]
2. Empty Set as a Subset:
- The empty set \( \emptyset \) is a subset of every set:
\[
\emptyset \subseteq A
\]
3. Subset of a Universal Set:
- Every set is a subset of the universal set \( U \), which contains all elements in a given context:
\[
A \subseteq U
\]
4. Number of Subsets:
- If a set \( A \) has \( n \) elements, then the number of subsets of \( A \) is \( 2^n \), including the empty set and \( A \) itself.
5. Number of Proper Subsets:
- A proper subset excludes the set itself. So, if \( A \) has \( n \) elements, the number of proper subsets of \( A \) is \( 2^n - 1 \).
---
Superset
A superset is the opposite of a subset. If all elements of \( A \) are also elements of \( B \), then \( B \) is considered a superset of \( A \).
Notation
- \( B \supseteq A \) denotes that \( B \) is a superset of \( A \).
- If \( B \supseteq A \) and \( B \neq A \), then \( B \) is called a proper superset of \( A \), written as \( B \supset A \).
Properties of Supersets
1. Reflexive Property:
- Every set is a superset of itself:
\[
A \supseteq A
\]
2. Universal Set as a Superset:
- The universal set \( U \) is a superset of every set within the context of that universe:
\[
U \supseteq A
\]
3. Empty Set:
- Every set is a superset of the empty set:
\[
A \supseteq \emptyset
\]
---
Summary of Formulas and Key Concepts
- Union of Two Sets: \( |A \cup B| = |A| + |B| - |A \cap B| \)
- Union of Three Sets: \( |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C| \)
- Intersection: \( A \cap B \) includes elements common to both sets.
- Complement: \( A^c = \{ x \mid x \in U \text{ and } x \notin A \} \)
- Subset: \( A \subseteq B \) means all elements of \( A \) are in \( B \).
- Superset: \( B \supseteq A \) means all elements of \( A \) are in \( B \).
Venn Diagram
A Venn Diagram is a visual representation used to illustrate the relationships between different sets. It shows how sets intersect, overlap, and relate to each other within a universal set. Venn diagrams are particularly helpful in understanding concepts like union, intersection, complement, and difference of sets.
Basic Structure of a Venn Diagram
- The entire rectangle typically represents the universal set \( U \).
- Each circle or closed shape inside the rectangle represents a set (e.g., \( A \), \( B \), \( C \)).
- Overlapping regions between circles represent the intersection of sets, where elements are common to multiple sets.
- Non-overlapping regions represent elements that belong only to one set.
---
Types of Venn Diagrams
1. Two-Set Venn Diagram
- Represents the relationship between two sets \( A \) and \( B \).
- Shows four distinct regions: elements in \( A \) only, elements in \( B \) only, elements in both \( A \) and \( B \) (intersection), and elements outside both \( A \) and \( B \) (complement within \( U \)).
2. Three-Set Venn Diagram
- Represents the relationships among three sets \( A \), \( B \), and \( C \).
- Shows multiple distinct regions, including those where elements belong to one, two, or all three sets, as well as the complement region (elements not in any of the sets).
3. General \( n \)-Set Venn Diagram
- Represents relationships among \( n \) sets.
- Becomes increasingly complex as the number of sets increases, but each region continues to represent specific combinations of intersections.
---
Key Concepts and Regions in a Venn Diagram
1. Union (\( A \cup B \)):
- The area covering all elements that are in \( A \), \( B \), or both.
- Visually, this is the entire area of both circles.
2. Intersection (\( A \cap B \)):
- The overlapping area where both \( A \) and \( B \) intersect.
- Represents elements common to both \( A \) and \( B \).
3. Difference (\( A - B \) or \( B - A \)):
- The area representing elements that are in \( A \) but not in \( B \) (for \( A - B \)), or in \( B \) but not in \( A \) (for \( B - A \)).
4. Complement (\( A^c \)):
- The area outside set \( A \) but within the universal set \( U \).
- Represents elements not in \( A \).
5. Symmetric Difference (\( A \triangle B \)):
- The area covering elements in \( A \) or \( B \) but not in both.
- Represents elements unique to each set, excluding the intersection.
---
Formulas for Venn Diagrams
1. Union of Sets
- For Two Sets:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
- For Three Sets:
\[
|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C|
\]
- For \( n \) Sets (Inclusion-Exclusion Principle):
\[
\left| \bigcup_{i=1}^n A_i \right| = \sum_{k=1}^n (-1)^{k+1} \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq n} \left| A_{i_1} \cap A_{i_2} \cap \dots \cap A_{i_k} \right|
\]
2. Intersection of Sets
- Intersection of Two Sets:
\[
|A \cap B|
\]
- Intersection of Three Sets:
\[
|A \cap B \cap C|
\]
- General Intersection:
\[
\bigcap_{i=1}^n A_i
\]
- There isn’t a specific formula for intersection beyond counting the number of elements common to all sets.
3. Difference of Sets
- Difference of Two Sets (\( A - B \)):
\[
|A - B| = |A| - |A \cap B|
\]
- Represents elements in \( A \) but not in \( B \).
4. Complement of a Set
- Complement of Set \( A \) within \( U \):
\[
|A^c| = |U| - |A|
\]
- This is the number of elements in \( U \) that are not in \( A \).
5. Symmetric Difference
- Symmetric Difference of Two Sets (\( A \triangle B \)):
\[
|A \triangle B| = |A| + |B| - 2|A \cap B|
\]
- Represents elements in \( A \) or \( B \) but not in both.
---
Examples Using Venn Diagram Formulas
1. Union Example:
- Let \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \).
- \( |A| = 3 \), \( |B| = 3 \), and \( |A \cap B| = 1 \) (element 3).
- Using the union formula:
\[
|A \cup B| = |A| + |B| - |A \cap B| = 3 + 3 - 1 = 5
\]
2. Intersection Example:
- Let \( A = \{a, b, c\} \) and \( B = \{b, c, d\} \).
- The intersection \( A \cap B = \{b, c\} \), so \( |A \cap B| = 2 \).
3. Difference Example:
- For sets \( A = \{1, 2, 3\} \) and \( B = \{3, 4\} \):
\[
|A - B| = |A| - |A \cap B| = 3 - 1 = 2
\]
- Elements in \( A - B \) are \( \{1, 2\} \).
4. Complement Example:
- If the universal set \( U = \{1, 2, 3, 4, 5\} \) and \( A = \{1, 2, 3\} \):
\[
|A^c| = |U| - |A| = 5 - 3 = 2
\]
- The elements in \( A^c \) are \( \{4, 5\} \).
5. Symmetric Difference Example:
- For sets \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \):
\[
|A \triangle B| = |A| + |B| - 2|A \cap B| = 3 + 3 - 2 \cdot 1 = 4
\]
- Elements in \( A \triangle B \) are \( \{1, 2, 4, 5\} \).
---
Summary of Key Venn Diagram Concepts and Formulas
- Union (\( A \cup B \)): Combines all elements from multiple sets, with the formula \( |A \cup B| = |A| + |B| - |A \cap B| \) for two sets.
- Intersection (\( A \cap B \)): Finds elements common to all sets involved, with notation \( A \cap B \) for two sets.
- Difference (\( A - B \)): Finds elements in one set but not in another, with the formula \( |A - B| = |A| - |A \cap B| \).
- Complement (\( A^c \)): Finds elements outside a set, with the formula \( |A^c| = |U| - |A| \).
- Symmetric Difference (\( A \triangle B \)): Includes elements in either set but not in both, with the formula \( |A \triangle B| = |A| + |B| - 2|A \cap B| \).
Venn Diagram
A Venn Diagram is a visual representation used to illustrate the relationships between different sets. It shows how sets intersect, overlap, and relate to each other within a universal set. Venn diagrams are particularly helpful in understanding concepts like union, intersection, complement, and difference of sets.
Basic Structure of a Venn Diagram
- The entire rectangle typically represents the universal set \( U \).
- Each circle or closed shape inside the rectangle represents a set (e.g., \( A \), \( B \), \( C \)).
- Overlapping regions between circles represent the intersection of sets, where elements are common to multiple sets.
- Non-overlapping regions represent elements that belong only to one set.
---
Types of Venn Diagrams
1. Two-Set Venn Diagram
- Represents the relationship between two sets \( A \) and \( B \).
- Shows four distinct regions: elements in \( A \) only, elements in \( B \) only, elements in both \( A \) and \( B \) (intersection), and elements outside both \( A \) and \( B \) (complement within \( U \)).
2. Three-Set Venn Diagram
- Represents the relationships among three sets \( A \), \( B \), and \( C \).
- Shows multiple distinct regions, including those where elements belong to one, two, or all three sets, as well as the complement region (elements not in any of the sets).
3. General \( n \)-Set Venn Diagram
- Represents relationships among \( n \) sets.
- Becomes increasingly complex as the number of sets increases, but each region continues to represent specific combinations of intersections.
---
Key Concepts and Regions in a Venn Diagram
1. Union (\( A \cup B \)):
- The area covering all elements that are in \( A \), \( B \), or both.
- Visually, this is the entire area of both circles.
2. Intersection (\( A \cap B \)):
- The overlapping area where both \( A \) and \( B \) intersect.
- Represents elements common to both \( A \) and \( B \).
3. Difference (\( A - B \) or \( B - A \)):
- The area representing elements that are in \( A \) but not in \( B \) (for \( A - B \)), or in \( B \) but not in \( A \) (for \( B - A \)).
4. Complement (\( A^c \)):
- The area outside set \( A \) but within the universal set \( U \).
- Represents elements not in \( A \).
5. Symmetric Difference (\( A \triangle B \)):
- The area covering elements in \( A \) or \( B \) but not in both.
- Represents elements unique to each set, excluding the intersection.
---
Formulas for Venn Diagrams
1. Union of Sets
- For Two Sets:
\[
|A \cup B| = |A| + |B| - |A \cap B|
\]
- For Three Sets:
\[
|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C|
\]
- For \( n \) Sets (Inclusion-Exclusion Principle):
\[
\left| \bigcup_{i=1}^n A_i \right| = \sum_{k=1}^n (-1)^{k+1} \sum_{1 \leq i_1 < i_2 < \dots < i_k \leq n} \left| A_{i_1} \cap A_{i_2} \cap \dots \cap A_{i_k} \right|
\]
2. Intersection of Sets
- Intersection of Two Sets:
\[
|A \cap B|
\]
- Intersection of Three Sets:
\[
|A \cap B \cap C|
\]
- General Intersection:
\[
\bigcap_{i=1}^n A_i
\]
- There isn’t a specific formula for intersection beyond counting the number of elements common to all sets.
3. Difference of Sets
- Difference of Two Sets (\( A - B \)):
\[
|A - B| = |A| - |A \cap B|
\]
- Represents elements in \( A \) but not in \( B \).
4. Complement of a Set
- Complement of Set \( A \) within \( U \):
\[
|A^c| = |U| - |A|
\]
- This is the number of elements in \( U \) that are not in \( A \).
5. Symmetric Difference
- Symmetric Difference of Two Sets (\( A \triangle B \)):
\[
|A \triangle B| = |A| + |B| - 2|A \cap B|
\]
- Represents elements in \( A \) or \( B \) but not in both.
---
Examples Using Venn Diagram Formulas
1. Union Example:
- Let \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \).
- \( |A| = 3 \), \( |B| = 3 \), and \( |A \cap B| = 1 \) (element 3).
- Using the union formula:
\[
|A \cup B| = |A| + |B| - |A \cap B| = 3 + 3 - 1 = 5
\]
2. Intersection Example:
- Let \( A = \{a, b, c\} \) and \( B = \{b, c, d\} \).
- The intersection \( A \cap B = \{b, c\} \), so \( |A \cap B| = 2 \).
3. Difference Example:
- For sets \( A = \{1, 2, 3\} \) and \( B = \{3, 4\} \):
\[
|A - B| = |A| - |A \cap B| = 3 - 1 = 2
\]
- Elements in \( A - B \) are \( \{1, 2\} \).
4. Complement Example:
- If the universal set \( U = \{1, 2, 3, 4, 5\} \) and \( A = \{1, 2, 3\} \):
\[
|A^c| = |U| - |A| = 5 - 3 = 2
\]
- The elements in \( A^c \) are \( \{4, 5\} \).
5. Symmetric Difference Example:
- For sets \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \):
\[
|A \triangle B| = |A| + |B| - 2|A \cap B| = 3 + 3 - 2 \cdot 1 = 4
\]
- Elements in \( A \triangle B \) are \( \{1, 2, 4, 5\} \).
---
Summary of Key Venn Diagram Concepts and Formulas
- Union (\( A \cup B \)): Combines all elements from multiple sets, with the formula \( |A \cup B| = |A| + |B| - |A \cap B| \) for two sets.
- Intersection (\( A \cap B \)): Finds elements common to all sets involved, with notation \( A \cap B \) for two sets.
- Difference (\( A - B \)): Finds elements in one set but not in another, with the formula \( |A - B| = |A| - |A \cap B| \).
- Complement (\( A^c \)): Finds elements outside a set, with the formula \( |A^c| = |U| - |A| \).
- Symmetric Difference (\( A \triangle B \)): Includes elements in either set but not in both, with the formula \( |A \triangle B| = |A| + |B| - 2|A \cap B| \).