# Let $A={a,b,c}$ and the relation $R$ be defined on $A$ as follows: $R= {(a,a)(b,c)(a,b)}$. Write the minimum number of ordered pairs to be ordered pairs to be added in $R$ to make $R$ reflexive and transitive.

$\begin{array}{1 1} 3 \\ 2 \\ 1 \\0 \end{array}$

Toolbox:
• A relation R in a set A is called $\mathbf{ reflexive},$ if $(a,a) \in R\;$ for every $\; a\in\;A$
• A relation R in a set A is called $\mathbf{symmetric}$, if $(a_1,a_2) \in R\;\Rightarrow\; (a_2,a_1)\in R \; for \;a_1,a_2 \in A$
• A relation R in a set A is called $\mathbf{transitive},$ if $(a_1,a_2) \in R$ and $(a_2,a_3) \in R \; \Rightarrow \;(a_1,a_3)\in R$ for all$\; a_1,a_2,a_3 \in A$
Given Let $A=\{a,b,c\}$

$R=\{(a,a),(b,c),(a,b)\}$

If we add (b,b) and (c,c) in R the relation R becomes reflexive

If we add (a,c) it becomes transitive

$(a,b),(b,c) \in R =>(a,c) \in R$

Hence the minimum number of ordered pair to be added to R is (b,b),(c,c),(a,c) is 3

edited Mar 20, 2013 by meena.p