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]}

Symmetric Difference using Venn Diagram

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}.

Symmetric Difference Venn Diagram

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}

Symmetric Difference

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}

Symmetric Difference of Two Sets

Therefore, the shaded part of the Venn diagram represents P △ Q = {a, b, f, j, k}.

Set Theory

Sets

Representation of a Set

Types of Sets

Pairs of Sets

Subset

Practice Test on Sets and Subsets

Complement of a Set

Problems on Operation on Sets

Operations on Sets

Practice Test on Operations on Sets

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


7th Grade Math Problems

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