Test by Bikram | Mock GATE | Test 4 | Question: 37

Question

A relation $R = \left \{ \left ( x,y \right ) \mid x,y\in N, x=7-y \right \}$

Here, $R$ is:

• A.   Symmetric and Reflexive, but not Transitive.
• B.   Symmetric, Reflexive and Transitive.
• C.   Only Symmetric, but neither Reflexive nor Transitive.
• D.   Neither Symmetric, Reflexive nor Transitive.

Answer 1

$R$ $=$ $\left \{ \left ( x,y \right )| x\equiv 7-y \right \}$

condition is $x + y =7$

Relation will contain;

$$\left \{ ,\left ( 6,1 \right ),\left (2,5 \right ) ,\left ( 5,2 \right ), \left ( 3,4 \right ),\left ( 4,3 \right )\right \}$$

It has all reverse pairs so it is symmetric,

It has no self pairs, so it is not reflexive.

It is not transitive also because if $\left ( 3,4 \right )$ and $\left ( 4,3 \right )$ belongs to R then $\left ( 3,3 \right )$ must belong to R which is false in given relation...

So option $C$.

Comments

This relation contains (6,1). To be Symmetric (1,6) must be present also. But It is not. So It is not Symmetric.