Consider the following statements for the two non-empty sets $A$ and $B$ : (1) $(\mathrm{A} \cap \mathrm{B}) \cup(\mathrm{A} \cap \overline{\mathrm{B}}) \cup(\overline{\mathrm{A}} \cap \mathrm{B})=\mathrm{A} \cap \mathrm{B}$ (2) $(\mathrm{A} \cup(\overline{\mathrm{A}} \cap \overline{\mathrm{B}}))=\mathrm{A} \cup \mathrm{B}$ Which of the above statements is/are correct ?

• A

  1 only
• B

  2 only
• C

  Both 1 and 2
• Correct answer
  D

  Neither 1 nor 2

Solution

1. $(A \cap B) \cup(A \cap \bar{B}) \cup(\bar{A} \cap B)=A \cup B$ Let us draw Venn diagram.
$\therefore 1$ is correct. 2. $A \cup(\bar{A} \cap \bar{B})=A$
$\therefore 2$ is not correct. Correct option (1).

Hence, the correct answer is option: 'D'.