Question

A, B, C and D are four sets such that $A\cap B=C\cap D=\varphi .$ Consider the following :
1. $A\cup C$ and $B\cup D$ are always disjoint.
2. $A\cap C$ and $B\cap D$ are always disjoint.
Which of the above statements is/are correct?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

Answer: (B)

2 only

Given that:

$A\cap B=C\cap D=\varphi$

$1.(A\cup C)\cap (B\cup D)$

$=\left[\right(A\cup C)\cap B]\cup \left[\right(A\cup C)\cap D]$

$=\left[\right(A\cap B)\cup (B\cap C\left)\right]\cup \left[\right(D\cap A)\cup \varphi ]$

$=(B\cap C)\cup (D\cap A)$

$\therefore (A\cup C)\cap (B\cup D)$ is not a disjoint set.

$2.(A\cap C)\cap (B\cap D)$

$=\left[\right(A\cap C)\cap B]\cap \left[\right(A\cap C)\cap D]$

$=\left[\right(A\cap B)\cap C]\cap [A\cap (C\cap D\left)\right]$

$=[\varphi \cap C]\cap [A\cap \varphi ]=\varphi \cap \varphi =\varphi$

$\therefore (A\cap C)\cap (B\cap D)$ is a disjoint set.

Hence, B is the correct option.