Question

Given the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, add a minimum number of ordered pairs so that the enlarged relation is symmetric, transitive and reflexive.

Answer 1

R = {(1, 2), (2, 3)}

$$\begin{gathered} \mathrm{For}R\mathrm{to}\mathrm{be}\mathrm{reflexive}\mathrm{it}\mathrm{must}\mathrm{have}(1,1),(2,2),(3,3). \\ \mathrm{For}R\mathrm{to}\mathrm{be}\mathrm{symmetric},\mathrm{all}\mathrm{the}\mathrm{ordered}\mathrm{pairs}\mathrm{upon}\mathrm{interchanging}\mathrm{the}\mathrm{elements}\mathrm{must}\mathrm{be}\mathrm{present}\mathrm{in}R. \\ \mathrm{Therefore},R\mathrm{must}\mathrm{contain}\left(2,1\right)\mathrm{and}\left(3,2\right),\left(3,1\right),\left(1,3\right). \\ \mathrm{Finally},\mathrm{for}R\mathrm{to}\mathrm{be}\mathrm{transitive},\mathrm{it}\mathrm{must}\mathrm{contain}\left(1,3\right). \end{gathered}$$

Hence, the number of ordered pairs to be added to R is 7, i.e. (1, 1), (2, 2), (3, 3), (1, 3), (3, 1), (2, 1), (3, 2).