Question

Let $S = \\{ 1, 2, 3, \ldots, 10 \\}$. Suppose $M$ is the set of all the subsets of $S$, then the relation $R = \\{ (A, B): A \cap B \neq \phi; A, B \in M \\}$ is:

• A. symmetric and reflexive only
• B. reflexive only
• C. symmetric and transitive only
• D. symmetric only

Answer 1

Answer: (D)

symmetric only

Answer 2

Let’s analyze the properties of the relation $R$.

Step 1. Reflexivity: For reflexivity to hold, each subset $A$ in $M$ should satisfy $A \cap A \neq \emptyset$. Since $A \cap A = A$, $R$ would be reflexive if $A \neq \emptyset$ for every $A \in M$. However, the empty set $\emptyset \in M$ does not satisfy $\emptyset \cap \emptyset \neq \emptyset$, so $R$ is not reflexive.

Step 2. Symmetry: If $(A, B) \in R$, then $A \cap B \neq \emptyset$. This implies $B \cap A \neq \emptyset$, so $(B, A) \in R$. Therefore, $R$ is symmetric.

Step 3. Transitivity: Suppose $(A, B) \in R$ and $(B, C) \in R$, meaning $A \cap B \neq \emptyset$ and $B \cap C \neq \emptyset$. However, $A \cap C$ may still be empty, so $R$ is not transitive.

Thus, the relation $R$ is symmetric only.