Question

Let $A$ and $B$ be two non-empty subsets of a set $X$ such that $A$is not a subset of $B$, then

• A. $A$is always a number of the complement of $B$
• B. $B$ is always a subset of $A$
• C. $A$and $B$ are always disjoint
• D. $A$ and the complement of $B$ are always non–disjoint

Answer 1

The correct option is D

$A$ and the complement of $B$ are always non–disjoint

Determine the relationship between the sets $A$ and $B$.

Let $X=\left\{1,2,3,4,5\right\}$

Given $A,B\subset X$

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

So that $A$is not a subset of $B$

Explanation for the correct answer:

Option (D):

$A$ and the complement of $B$ are always non–disjoint.

Determine: $A\cap B^{\text{'}}$

$$\begin{gathered} B^{\text{'}}=\left\{2,3\right\} \\ A\cap B^{\text{'}}=\left\{2\right\} \end{gathered}$$

Therefore, $A$ and the complement of $B$ are always non–disjoint.

Explanation for wrong answer:

Option (A):

$A$is always a number of the complement of $B$

$B^{\text{'}}=\left\{2,3\right\}$

$A=\left\{1,2\right\}$

$A\ne B\text{'}$

Therefore, option (A) is incorrect.

Option (B):

$B$is always a subset of $A$

$\left(B=\left\{1,4,5\right\}\right)\not\subset \left(A=\left\{1,2\right\}\right)$

Therefore, option (B) is incorrect.

Option (C):

$A$ and $B$ are always disjoint.

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

Therefore, option (C) is incorrect.

Hence, option (D) is the correct answer.