Complement of a Set: Overview, Questions, Preparation

Sets are representations of well-defined elements and do not vary with perspectives. It is typically represented by a letter.

The complement of a set includes all the elements which are NOT in the given set.

In set-builder notation, the complement of set A is represented as: A' = {x ∈ U: x ∉ A}, where U is the universal set.

For example,if set A =(1,3,5,7,...)

The complement of the set A is A’ =(2,4,6,8,..)

Things to Remember

The complement of an empty set is a universal set.

∅' = U

The complement of a universal set is an empty set. 

U' = ∅

Importance and Weightage

This topic comes all along with the mathematical curriculum. Students are introduced to the concept in class 11 but find its application in subsequent classes.

1. Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 3, 4}, B = { 2, 4, 6, 8 } and C = { 3, 4, 5, 6 }. Find (i) A′ (ii) B′ (iii) (A ∪ C)′ (iv) (A ∪ B)′ (v) (A′)′ 

Solution.

(i) A’ = {5,6,7,8,9}

(ii) B’= {1,3,5,7,9}

(iii) (AuC)’= {1,2,7,8,9}

(iv) (A u B)’= {5,7,9}

(v) (A’)’= {1, 2, 3, 4}

2. If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets : (i) A = {a, b, c} (ii) B = {d, e, f, g} (iii) C = {a, c, e, g} (iv) D = { f, g, h, a}

Solution.

(i) A’ = {d,e,f,g,h}

(ii) B’ = {a,b,c,h}

(iii) C’ = {a,b,d,f,h}

(iv) D’ = {b,c,d,e}

3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:

(i) {x : x is an even natural number}

(ii) { x : x is an odd natural number }

(iii) {x : x is a positive multiple of 3}

(iv) { x : x is a prime number }

(v) {x : x is a natural number divisible by 3 and 5}

Solution.

(i) {x: x is an even natural number}´ = {x: x is an odd natural number}

(ii) {x: x is an odd natural number}´ = {x: x is an even natural number}

(iii) {x: x is a positive multiple of 3}´ = {x: x ∈ N and x is not a multiple of 3}

(iv) {x: x is a prime number}´ ={x: x is a positive composite number and x = 1}

(v) {x: x is a natural number divisible by 3 and 5}´ = {x: x is a natural number that is indivisible by 3 or 5}

Q: What is the definition of a set?

Q: What is a universal set?

Q: What is the complement of a set?

Q: How do I represent sets?

Q: What are the basic operations involving sets?