Question

Let $A$and $B$ be two sets. Then

• A. $A\cup B\subseteq A\cap B$
• B. $A\cap B\subseteq A\cup B$
• C. $A\cap B=A\cup B$
• D. None of these

Answer 1

The correct option is B

$A\cap B\subseteq A\cup B$

Determine the relation between set $A$and $B$

Let $A=\left\{1,2\right\}\text{and}B=\left\{1,4,5\right\}$

$A\cap B=\left\{1\right\}$

$A\cup B=\left\{1,2,4,5\right\}$

Explanation for correct option

Option (B):

To check if $A\cap B\subseteq A\cup B$

$$\begin{gathered} LHS=\left\{1\right\} \\ RHS=\left\{1,2,4,5\right\} \\ \Rightarrow LHS\subseteq RHS \end{gathered}$$

Therefore, option (B) is correct.

Explanation for incorrect option

Option (A):

To check if $A\cup B\subseteq A\cap B$

$$\begin{gathered} RHS=\left\{1\right\} \\ LHS=\left\{1,2,4,5\right\} \\ \Rightarrow LHS⊈RHS \end{gathered}$$

Therefore, option (A) is incorrect.

Option (C):

To check if $A\cap B=A\cup B$

$$\begin{gathered} LHS=\left\{1\right\} \\ RHS=\left\{1,2,4,5\right\} \\ \Rightarrow LHS\ne RHS \end{gathered}$$

Therefore, option (C) is incorrect.

Option (D): None of these

Option (B) is correct, Therefore, option (D) becomes invalid.

Hence, option (B) is the correct answer.