A relation R in the set \(A = \{1,2,3,4\}\) is given by \(R = \{(1,1), (2,2), (1,2), (2,3), (3,4), (4,4), (1,3), (2,4), (1,4)\}\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Transitive
Reflexive: no, since \((3, 3) \notin R\).
Symmetric: no, since \((1, 2) \in R\) but \((2, 1) \notin R\).
Transitive: yes. Pairs with a common middle element (other than the pairs \((a, a)\), which give nothing new) and the pair they need:
\((1, 2), (2, 3) \Rightarrow (1, 3) \in R\)
\((1, 2), (2, 4) \Rightarrow (1, 4) \in R\)
\((1, 3), (3, 4) \Rightarrow (1, 4) \in R\)
\((2, 3), (3, 4) \Rightarrow (2, 4) \in R\)
Every required pair is in R, so R is transitive but not reflexive, which also rules out Option (4).