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 and B are always disjoint
• B. A is always a subset of A
• C. A and the complement of B are always disjoint
• D. A is never equal to the complement of B

Answer 1

Answer: (B)

A is always a subset of A

Answer 2

The condition $A \not\subset B$ implies that there exists at least one element $x$ such that $x \in A$ and $x \notin B$.

Let's analyze each option:

1. A and B are always disjoint: If A and B are always disjoint, then $A \cap B = \emptyset$. Consider a counterexample: Let $X = \{1, 2, 3\}$, $A = \{1, 2\}$, and $B = \{2, 3\}$. Here, A is not a subset of B because $1 \in A$ but $1 \notin B$. However, $A \cap B = \{2\}$, which is not an empty set. Therefore, A and B are not always disjoint. This option is incorrect.

2. A is always a subset of A: This statement, $A \subseteq A$, is a fundamental property of set theory. Every set is a subset of itself. This statement is always true, regardless of the relationship between A and B or whether A is a non-empty subset of X. Since the question asks which statement is true given the condition, and this statement is universally true, it is a correct option.

3. A and the complement of B are always disjoint: If A and the complement of B ($B^c$) are always disjoint, then $A \cap B^c = \emptyset$. The condition $A \not\subset B$ means there exists an element $x \in A$ such that $x \notin B$. If $x \notin B$, then by definition of complement, $x \in B^c$. So, there exists an element $x$ such that $x \in A$ and $x \in B^c$. This implies that $A \cap B^c \neq \emptyset$. Therefore, A and the complement of B are always non-disjoint. The statement that they are always disjoint is incorrect.

4. A is never equal to the complement of B: This statement claims $A \neq B^c$. Consider a counterexample: Let $X = \{1, 2, 3, 4\}$, $A = \{1, 2\}$, and $B = \{3, 4\}$. Here, A is not a subset of B because $1 \in A$ but $1 \notin B$. The complement of B is $B^c = X - B = \{1, 2\}$. In this example, $A = B^c$. Since we found a case where A is equal to the complement of B, the statement "A is never equal to the complement of B" is incorrect.

Based on the analysis, options A, C, and D are false statements. Option B is a universally true statement in set theory, and thus it is true under the given conditions.