Question

Mark the correct alternative of the following.
The symmetric difference of $ A $ and $ B $ is not equal to ?
(A) $ \left( {A - B} \right) \cap \left( {B - A} \right) $
(B) $ \left( {A - B} \right) \cup \left( {B - A} \right) $
(C) $ \left( {A \cup B} \right) - \left( {A \cap B} \right) $
(D) $ \{ (A \cup B) - A\} \cup \{ (A \cup B) - B\} $

Answer 1

Hint: Symmetric difference of two sets A and B is the union of difference of sets A and B. That is $ A - B $ or $ B - A $. We will see all possible cases and then finally arrive at the conclusion on the basis of definition.

Complete step-by-step answer:
Symmetric difference, $ A\Delta B $ of two sets, A and B is given by
 $ A\Delta B = \left( {A - B} \right) \cup \left( {B - A} \right) $
By Venn Diagram, it can be represented as

Clearly, by definition, option (B) satisfies the condition of symmetric difference.
 $ A - B $ is the collection of elements that are in A but not in B.
 $ \Rightarrow A - B = \{ x:x \in A $ but $ x \notin B\} $
By Venn diagram we can represent it as

 $ B - A $ is the collection of elements that are in B but not in A.
 $ \Rightarrow B - A = \{ x:x \in B $ but $ x \notin A\} $
By Venn diagram we can represent it as

In the two diagrams, we can clearly observe that there is no element common in $ A - B $ and $ B - A $
 $ \Rightarrow \left( {A - B} \right) \cap \left( {B - A} \right) = \phi $
Therefore, $ \left( {A - B} \right) \cap \left( {B - A} \right) $ is not a symmetric difference.
 $ \left( {A \cup B} \right) $ is the collection of elements that are either in A or in B or in both.
 $ \Rightarrow A \cup B = \{ x:x \in A $ or $ x \in B\} $
By Venn diagram we can represent it as

 $ A \cap B $ is the collection of elements that are in both A and B.
 $ \Rightarrow A \cap B = \{ x:x \in A $ and $ x \in B\} $
By Venn diagram we can represent it as

Therefore, by observing above two diagrams we can conclude that
By Venn diagram, $ \left( {A \cup B} \right) - \left( {A \cap B} \right) $ can be represented as

Which is exactly like the Venn diagram that represents symmetric difference.
 $ (A \cup B) - A $ is the collection of elements that are in $ A \cup B $ but not in A.
 $ \Rightarrow (A \cup B) - A = \{ x:A \cup B $ but $ x \notin A\} $
By Venn diagram it can be represented as

 $ (A \cup B) - B $ is the collection of elements that are in $ A \cup B $ but not in B.
By Venn diagram it can be represented as

By observing above two diagrams we can conclude that
 $ \{ (A \cup B) - A\} \cup \{ (A \cup B) - B\} $ can be represented by Venn diagram as

Which is the same representation as symmetric difference.
Therefore, by above explanation, the correct answer is, option (A) $ \left( {A - B} \right) \cap \left( {B - A} \right) $

So, the correct answer is “Option A”.

Note: From the above question, you can understand the importance of Venn diagram. This question would have been difficult to explain in theory. Diagrammatic representation helped to solve it easily.