Question

Consider the following statements on a set $A = \{1, 2, 3\}$

$S_1: R = \{(1, 1), (2, 2)\}$ is a reflexive relation on $A$

$S_2: R = \{(3, 3)\}$ is symmetric and transitive but not reflexive on $A$

Which of the following statement on set $A$ is true?

• A. $S_1$ only
• B. $S_2$ only
• C. Both $S_1$ and $S_2$
• D. Neither $S_1$ nor $S_2$

Answer 1

Answer: (B)

$S_2$ only

Answer 2

For a relation $R$ on $A = \{1, 2, 3\}$ to be reflexive, it must contain $(1, 1)$, $(2, 2)$, and $(3, 3)$.

• $S_1: R = \{(1, 1), (2, 2)\}$

  • Reflexivity: Fails because $(3, 3)$ is missing.

• $S_2: R = \{(3, 3)\}$

  • Symmetry: Holds (only one pair, and symmetric by default).
  • Transitivity: Holds (only one pair, so the condition is trivially true).
  • Reflexivity: Fails because $(1, 1)$ and $(2, 2)$ are missing.

Thus, only $S_2$ correctly describes a relation on $A$ that is symmetric and transitive but not reflexive.