Sets (Union, Intersection, Complement..) with Examples

A set is a collection of objects called members. For example, pens, cars, people. In mathematical terms, objects are called elements

A set can be defined by listing the elements, e.g {2,4,6} or using descriptions e.g A = {all multiples of 3} or using Venn Diagrams.

Terms used in sets

Universal set

The universal set contains all elements. S = {people in a building}

Empty Set (∅)

Set without any elements

Element of (∈)

For example A={2,4,6}, therefore 6 ∈ A

Not Element of (∉)

Referring above; 7 ∉ A

Union (∪)

Union means combine all 2 or more set elements.
A={1, 3, 5} and B={2, 4, 6}. A ∪B = {1, 2, 3, 4, 5, 6}. NB: Do not repeat elements. For example the union of {1, 3 ,5} and {1, 3, 4} = {1,3,4,5}. This is wrong; {1, 1, 3, 3, 4, 5}.

Intersection (∩)

Combine (only) elements found in both 2 or more sets.
A={1,3,5} and B= {1, 3, 4} Intersection of A and B is A ∩ B = {1,3}

Compliment set A′

Elements outside a set
S ={1, 3, 5, 7} and A={1,3}. A′ = {5,7}

Example 1

    ξ = {x : x is a real number: x<30}
    A = {x: x is a prime number}
    B = {x : x is a multiple of 3}
    C = {x : x is a multiple of 4}
    Find
    i) A∩B
    ii) A∩C
    iii) B∩C
    A = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
    B= {3, 6, 9, 12, 15, 18, 21, 24, 27}
    C= {4, 8, 12, 16, 20, 24, 28}
    i) A∩B = {3}
    ii) A∩C = {0} /∅
    iii) B∩C = {12, 24}
  

Example 2

NB: In dealing with compliments first list all elements in the universal set

    ξ = {x is an integer : –2 ≤ x ≤ 5}
    P = {x : –2 < x < 3}
    Q = {x : 0 < x ≤ 4}
    |*Solution*|
    Universal set = {-2, -1, 0, 1, 2, 3, 4, 5}
    P = {-1, 0, 1, 2}
    Q = {1, 2, 3, 4}

    P∩Q = {1,2}

    P′ = {-2, 3, 4, 5}

    P∪Q = {-1, 0, 1, 2, 3, 4}
  

Example 3 (ZIMSEC, Nov 2010; q2)

 Shade
    i) A′∪B
    ii) A∩B′∩C
  

Example 4 (ZIMSEC Nov 2019; q4)

    S = {x:-3 ≤ x ≤ 3; x is an integer}
    A = {x:-2 ≤ x  < 2}
    B= {x:-1 ≤x ≤3}
    Find
    i)1. A
    i)2. A′∪B′
    ii) n(A∩B)
    Solutions
    S = {-3, -2, -1, 0, 1, 2, 3}

    i)1. A = {-2, -1, 0, 1}
      B = { -1, 0, 1, 2, 3 }

    i)2. A′∪B′
    A′ = {-3, 2, 3}
    B′ = {-3, -2}
    A′∪B′ = {-3, -2, 2, 3}

    ii) n(A∩B)  = {-1, 0, 1} = 3