Question

If \[A\] and \[B\]are disjoint sets then, \[A\vartriangle B = \]____________
A) \[A \cup B\]
B) \[A \cap B\]
C) \[A - B\]
D) \[B - A\]

Answer 1

Hint- In set theory, the symmetric difference of two sets is the set of the elements which are in either of the set and not in their intersection. The symmetric difference of two sets of A and B is denoted by \[A\vartriangle B\]. It is also known as disjunctive union.
In set theory, we can say that two sets are called disjoint sets if they have no element in common. The intersection of two disjoint sets is always empty or null set.

Complete step by step answer:
It is given that, \[A\] and \[B\]are disjoint sets. It means they do not have any common elements.
Let us consider, the given sets as,
\[A = \{ a,b,c\} \] and \[B = \{ p,q,r\} \]
They do not have any common elements as they are disjoint.
Since, \[A\] and \[B\]are disjoint sets, therefore their intersection is a null set.
That is, \[A \cap B = \phi \]
The elements of \[A\vartriangle B\] belong to \[A\] and \[B\] but do not belong to its intersection.
So, we get,
\[A\vartriangle B = A \cup B\]
Hence,
The correct option is (A) \[A \cup B\].

Note – The basic difference of symmetric difference and union of two sets that is, for the union of two sets we consider the common elements also. But for symmetric difference we do not consider the common elements.
Let us consider two sets be,
\[{\text{A}} = \left\{ {{\text{1}},{\text{2}},{\text{3}},{\text{4}},{\text{5}},{\text{6}},{\text{7}},{\text{8}},{\text{9}},{\text{1}}0} \right\}\]
And \[{\text{B}} = \left\{ {{\text{2}},{\text{4}},{\text{6}},{\text{8}},{\text{1}}0} \right\}\]
We have to find the union of the sets \[A\] and \[B\],
Then the union of these two set, \[A \cup B = \{ 1,2,3,4,5,6,7,8,9,10\} \]
And the symmetric difference of these two sets is \[A\vartriangle B = \{ 1,3,5,7,9\} \]
The symmetric difference of two sets is the subset of the union of those two sets.