Complement Of A Set | Complement Of A Universal Set

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Complement of a Set

In complement of a set if ξ be the universal set and A a subset of ξ, then the complement of A is the set of all elements of ξ which are not the elements of A. Symbolically, we denote the complement of A with respect to ξ as A’.

For Example; If ξ = {1, 2, 3, 4, 5, 6, 7}A = {1, 3, 7} find A'.Solution: We observe that 2, 4, 5, 6 are the only elements of ξ which do not belong to A.Therefore, A' = {2, 4, 5, 6}Note:

The complement of a universal set is an empty set.The complement of an empty set is a universal set.The set and its complement are disjoint sets.

For Example;

1. Let the set of natural numbers be the universal set and A is a set of even natural numbers, then A' {x: x is a set of odd natural numbers} 2. Let ξ = The set of letters in the English alphabet. A = The set of consonants in the English alphabet then A' = The set of vowels in the English alphabet. 3. Show that; (a) The complement of a universal set is an empty set. Let ξ denote the universal set, then ξ' = The set of those elements which are not in ξ. = empty set = ϕ Therefore, ξ = ϕ so the complement of a universal set is an empty set. (b) A set and its complement are disjoint sets. Let A be any set then A' = set of those elements of ξ which are not in A'. Let x ∉ A, then x is an element of ξ not contained in A' So x ∉ A' Therefore, A and A' are disjoint sets. Therefore, Set and its complement are disjoint sets

Similarly, in complement of a set when U be the universal set and A is a subset of U. Then the complement of A is the set all elements of U which are not the elements of A. Symbolically, we write A' to denote the complement of A with respect to U. Thus, A' = {x : x ∈ U and x ∉ A} Obviously A' = {U - A} For Example; Let U = {2, 4, 6, 8, 10, 12, 14, 16} A = {6, 10, 4, 16} A' = {2, 8, 12, 14} We observe that 2, 8, 12, 14 are the only elements of U which do not belong to A.

Some properties of complement sets

(i) A ∪ A' = A' ∪ A = ∪ (Complement law) (ii) (A ∩ B') = ϕ (Complement law) (iii) (A ∪ B) = A' ∩ B' (De Morgan’s law) (iv) (A ∩ B)' = A' ∪ B' (De Morgan’s law) (v) (A')' = A (Law of complementation) (vi) ϕ' = ∪ (Law of empty set (vii) ∪' = ϕ and universal set)

● Set Theory

● Sets

● Objects Form a Set

● Elements of a Set

● Properties of Sets

● Representation of a Set

● Different Notations in Sets

● Standard Sets of Numbers

● Types of Sets

● Pairs of Sets

● Subset

● Subsets of a Given Set

● Operations on Sets

● Union of Sets

● Intersection of Sets

● Difference of two Sets

● Complement of a Set

● Cardinal number of a set

● Cardinal Properties of Sets

● Venn Diagrams

7th Grade Math Problems

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