2.3: Universal Sets & Venn Diagrams
Universal Set and Venn Diagrams
Definition of Universal Set: A universal set is a general set that contains all elements under discussion.
John Venn: Developed Venn diagrams (1843 – 1923) to visually illustrate relationships among sets.
Representation: The universal set is represented by a rectangle, while subsets within the universal set are depicted by circles or ovals.
Basic Terminology Related to Sets
Disjoint Sets: Two sets that have no elements in common.
Proper Subsets: A set A is a proper subset of set B if all elements of set A are also elements of set B.
Equal Sets: Sets A and B are equal if A = B; this results in A being a subset of B (A ⊆ B) and B being a subset of A (B ⊆ A).
Sets with Common Elements: If sets A and B share at least one common element, the circles representing the sets must overlap in a Venn diagram.
Examples of Set Relationships
Example 1: Using a Venn Diagram
Universal Set U: U = {a, b, c, d, e, f, g}
Set B: B = {d, e}
Set A: To find:
Elements in A but not B: A - B = {a, b, c}
Elements in U but not B: U - B = {a, b, c, f, g}
Elements in both A and B: A ∩ B = {d}
Elements in A or B: A ∪ B = {a, b, c, d, e}
Example 2: Identifying Regions in the Venn Diagram
Regions Defined: The Roman numerals I, II, III, and IV in a Venn Diagram are not elements of the sets; they define various regions within the diagram.
Complement of a Set
Definition of Complement: The complement of set A, denoted A’, is defined as the set of all elements in the universal set that are not in A.
The complement can be represented in set-builder notation: A' = {x | x ∈ U and x ∉ A}.
Finding the Complement
For example:
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {1, 3, 4, 7}
To find A’: A' = {2, 5, 6, 8, 9}
Set Operations
Intersection of Sets
Definition: The intersection of sets A and B, denoted A ∩ B, is the set of elements that are common to both sets.
Expression in set-builder notation: A ∩ B = {x | x ∈ A and x ∈ B}.
Examples of Intersection
Given Set Examples:
a. {1, 3, 5, 7, 10} ∩ {6, 7, 10, 11} = {7, 10}
b. {1, 2, 3} ∩ {4, 5, 6, 7} = ∅
c. {1, 2, 3} ∩ {Ø} = ∅
Union of Sets
Definition: The union of sets A and B, denoted A ∪ B, is the set of all elements that are in either set A, set B, or both.
Expression in set-builder notation: A ∪ B = {x | x ∈ A or x ∈ B}.
Examples of Union
Given Set Examples:
a. {1, 3, 5, 7, 10} ∪ {6, 7, 10, 11} = {1, 3, 5, 6, 7, 10, 11}
b. {1, 2, 3} ∪ {4, 5, 6, 7} = {1, 2, 3, 4, 5, 6, 7}
c. {1, 2, 3} ∪ {Ø} = {1, 2, 3}
Empty Set in Intersection and Union
Properties:
For any set A: A ∩ Ø = Ø
For any set A: A ∪ Ø = A
Example:
If Set A = {a, b, c} and Set B = Ø:
{a, b, c} ∩ Ø = Ø
{a, b, c} ∪ Ø = {a, b, c}
Performing Set Operations
Multiple Set Operations: Some problems may involve more than one set operation. The order of operations must be followed
Operations inside parentheses are performed first, followed by complements, then intersections, and finally unions.
Complex Operation Example
For Given Sets U = {a, b, c, d, e}, A = {b, c}, B = {b, c, e}:
To find (A ∪ B)’:
First compute A ∪ B = {b, c, e} then A ∪ B’ = {a, d}.
Additional Set Check Points
Check Point 6
A' ∩ B' for given sets gives the results:
A' = {a, d, e}, B' = {a, d}
Intersection: A' ∩ B' = {a, d}.
Check Point 7
Use a Venn diagram to detemine specific sets, like:
A ∩ B = {5}
(A ∩ B)’ = {2,3,7,11,13,17,19}
A ∪ B = {2,3,5,7,11,13}
(A ∪ B)’ = A’ = {17,19}
Practical Use of Sets in English
Set Operations: Set operations and Venn diagrams provide precise methods for organizing, classifying, and describing various sets and subsets in daily life.
Logical Connectives:
“Or” refers to the union of sets.
“And” refers to the intersection of sets.
Cardinal Numbers and the Union of Two Finite Sets
Formula for the cardinal number of the union of two finite sets:
Example: Among U.S. presidents, 27 had dogs, 12 had cats, and 10 had both:
Thus, 29 presidents had either dogs or cats in the White House.