Question

What is the symmetric difference of sets A and B, where A = {1, 4, 9, 10, 15} and B = {2, 4, 7, 10} ?

Answer 1

Hint: We know that the symmetric difference of two sets, is the set that contains only those elements which are present either in any one of these sets, but not in both of them. We can also find the symmetric difference by subtracting the intersection from the union.

Complete step by step answer:
We know that the symmetric difference of two sets A and B, is the set of elements that contains only those elements which are present either in set A, or in set B, but not in both of them.
In terms of set theory, we can say that the symmetric difference of two sets is the difference of union and intersection of those two sets.
We must also remember that the symmetric difference of set A with respect to set B is represented as $A\oplus B$.
Thus, mathematically, we can also write that
$A\oplus B=A\bigcup B-A\bigcap B$
We can use Venn diagrams to represent the symmetric difference as follows,

Here in this question, we have A = {1, 4, 9, 10, 15} and B = {2, 4, 7, 10}.
We know that the union of two sets is a set containing all elements that are in A or in B.
So, $A\bigcup B$ = {1, 2, 4, 7, 9, 10, 15}.
We also know that the intersection of sets is a set containing all those elements which are present in both A and B.
So, $A\bigcap B$ = {4, 10}.
We know that $A\oplus B=A\bigcup B-A\bigcap B$.
So, $A\oplus B$ = {1, 2, 7, 9, 15}.
Thus, the symmetric difference of sets A and B is {1, 2, 7, 9, 15}.

Note: We must remember that symmetric difference is also known as disjunctive union. We must keep in mind that different texts use different symbols for symmetric difference. Some of these forms are $A\oplus B,A\mathrm{\Delta }B\text{ and }A\mathrm{\Theta }B$.