Question

If $A \cap B = \emptyset $, then $A\Delta B = $
${\text{A}}{\text{. A}} \cap {\text{B}}$
${\text{B}}{\text{. A}} \cup {\text{B}}$
${\text{C}}{\text{. A - B}}$
${\text{D}}{\text{. B - A}}$

Answer 1

Hint: $A\Delta B$ is called the symmetric difference between A and B and is defined as $(A - B) \cup (B - A)$. To solve the given question draw a Venn diagram for better understanding, also $A \cap B = \emptyset $ as given in question which means A and B have all elements different i.e there are no elements common between both A and B, use this to find $A\Delta B$.

Complete step-by-step answer:
Given in the question, $A \cap B = \emptyset $.
Drawing the Venn diagram for $A \cap B$.

Venn diagram for A – B,

Now, we know $A\Delta B = (A - B) \cup (B - A)$
Since, the intersection of A and B is $\emptyset $.
Also, (A – B) refers to the elements that belong to A alone and (B – A) refers to the elements that belong to B alone.
Now using the condition given in question, $A \cap B = \emptyset $, it implies that (A – B) = A and (B – A) = B.
Hence, $A\Delta B = (A - B) \cup (B - A) = A \cup B$.
Therefore, the correct option is option (B).

Note – Whenever such types of questions appear, always use the condition given in the question to solve. In this question we have used the properties of sets to get the answer to the given question. We have used the standard formula of symmetric difference, i.e. $A\Delta B = (A - B) \cup (B - A)$