Q.On the set $A = \{1, 2, 3\}$, examine whether $R = \{(1, 1), (2, 2), (1, 2)\}$ is reflexive, symmetric and transitive.

Check each property: reflexive needs all three self-pairs; symmetric needs every reverse; transitive needs every chain's shortcut. …

Test on $A = \{1, 2, 3\}$.

Reflexive? Requires $(1,1), (2,2), (3,3)$ all present. Here $(3, 3)$ is missing, so $R$ is not reflexive.

Symmetric? Requires the reverse of every pair. $(1, 2) \in R$ but $(2, 1) \notin R$, so $R$ is not symmetric.

Transitive? Requires $(a, c) \in R$ whenever $(a, b), (b, c) \in R$. The only chain to test is $(1, 1)$ and $(1, 2) \Rightarrow$ need $(1, 2)$ — present; and $(1, 2)$ and $(2, 2) \Rightarrow$ need $(1, 2)$ — present. No chain forces a missing pair, so $R$ **is transiti …

Students often overlook that reflexivity must cover every element of $A$ — the presence of $(1,1)$ and $(2,2)$ is not enough while $(3,3)$ is absent. For transitivity, remember a propert …