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# Symmetric Difference using Venn Diagram

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# Symmetric Difference using Venn Diagram

The symmetric difference using Venn diagram of two subsets A and B is a sub set of U, denoted by A △ B and is defined by

A B = (A – B) ∪ (B – A)

Let A and B are two sets. The symmetric difference of two sets A and B is the set (A – B) ∪ (B – A) and is denoted by A △ B.

Thus, A B = (A – B) ∪ (B – A) = {x : x A ∩ B}

or, A B = {x : [x ∈ A and x ∉ B] or [x ∈ B and x ∉ A]} The shaded part of the given Venn diagram represents  B.

A △ B is the set of all those elements which belongs either to A or to B but not to both.

A △ B is also expressed by (A ∪ B) – (B ∩ A).

It follows that A △ ∅ = A for all subset A,

A △ A = ∅ for all subset A

Properties of symmetric difference:

(i) A △ B = B △ A;                                       [Commutative property]

(ii) A △ (B △ C) = (A △ B) △ C                       [Associative property]

Example to find the symmetric difference using Venn diagram:

1. If A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {1, 3, 5, 6, 7, 8, 9}, then A – B = {2, 4}, B – A = {9} and A B = {2, 4, 9}. Therefore, the shaded part of the Venn diagram represents A △ B = {2, 4, 9}.

2. If A = {1, 2, 4, 7, 9} and B = {2, 3, 7, 8, 9} then A △ B = {1, 3, 4, 8} Therefore, the shaded part of the Venn diagram represents A △ B = {1, 3, 4, 8}.

3. If P = {a, c, f, m, n} and Q = {b, c, m, n, j, k} then P △ Q = {a, b, f, j, k} Therefore, the shaded part of the Venn diagram represents P △ Q = {a, b, f, j, k}.

Set Theory

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Types of Sets

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Problems on Operation on Sets

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Word Problems on Sets

Venn Diagrams

Venn Diagrams in Different Situations

Relationship in Sets using Venn Diagram

Examples on Venn Diagram

Practice Test on Venn Diagrams

Cardinal Properties of Sets

Symmetric Difference using Venn Diagram

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