Symmetric Difference of Two Sets

Let A and B be any two sets. The symmetric difference of A and B is the set (A–B)∪(B–A). It is denoted by A∆B and read as A symmetric difference B. The symbol ‘∆’ is
used to denote the symmetric difference.

∴A∆B=(A–B)∪(B–A)
={x:x∈A or x∈B but x∉A∩B}
The symmetric difference of sets A and B is represented by the following Venn diagram

Hence, the shaded portion represents the symmetric difference of sets A and B.

[Definition] The symmetric difference of two sets S and T is the set of objects that are in one and only one of the sets. The symmetric difference is written S∆T. In curly brace notation:

S∆T={(S–T)∪(T–S)}

📎 Example 1. Let C={1,2,3,4} and D={3,4,5,6}
Now, C–D={1,2}

D–C={5,6}
∴C∆D=(C–D)∪(D–C)={1,2,5,6}

📎 Example 2: Symmetric differences
Let E be the set of non-negative multiples of two that are no more than twenty four. Let F be the nonnegative multiples of three that are no more than twenty four. Then

E∆F={2,3,4,8,9,10,14,15,16,20,21,22}
Another way to think about this is that we need numbers that are positive multiples of 2 or 3 (but not both) such as {6,12,18,24,…} that are no more than 24.

🍫 Method of Finding Symmetric Difference of Two Sets
If two sets C and D (say) are given, then we find their symmetric difference with the help of following steps
Step I. Write the given sets in tabular form (if not given in tabular form) and assume them C and D (say).
Step II. Find the difference of sets C and D i.e., C–D.
Step III. Find the difference of sets D and C i.e., D–C.
Step IV. Find the union of sets obtained from steps II and III, which will give the required symmetric difference.

Worked out Problem:
Find the symmetric difference of sets C={1,3,5,6,7} and D={3,7,8,9}.
Step I. Write the given sets in tabular form.

Given sets are C={1,3,5,6,7} and D={3,7,8,9} which are in tabular form.

Step II. Find the difference of sets C and D i.e., C–D.

We know that, difference of sets C and D is the set of those elements of C, which are not present in D. So, C–D={1,3,5,6,7}-{3,7,8,9}

Step III. Find the difference of sets D and C i.e., D–C.

We know that, difference of sets D and C is the set of
those elements of D, which are not present in C.
So, D–C={3,7,8,9}−{1,3,5,6,7}={8,9}

Step IV. Find the union of sets obtained from steps II and III, which will give the required symmetric difference.

∴ Required symmetric difference,

C∆D=(C–D)∪(D–C)={1,5,6}∪{8,9}={1,5,6,8,9}