Question

Let R be the relation on the set A = {1, 2, 3, 4} given by
R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Then,
(a) R is reflexive and symmetric but not transitive
(b) R is reflexive and transitive but not symmetric
(c) R is symmetric and transitive but not reflexive
(d) R is an equivalence relation

Answer 1

(b) R is reflexive and transitive but not symmetric.

$$\begin{gathered} \mathrm{Reflexivity}:\mathrm{Clearly},(a,a)\in R\forall a\in A \\ \mathrm{So},R\mathrm{is}\mathrm{reflexive}\mathrm{on}A. \\ \mathrm{Symmetry}:\mathrm{Since}\left(1,2\right)\in R,\mathrm{but}\left(2,1\right)\notin R, \\ R\mathrm{is}\mathrm{not}\mathrm{symmetric}\mathrm{on}A. \\ \mathrm{Transitivity}:S\mathrm{ince},\left(1,3\right),\left(3,2\right)\in R\mathrm{and}\left(1,2\right)\in R, \\ R\mathrm{is}\mathrm{transitive}\mathrm{on}A. \end{gathered}$$