Question

If R = {(a, b) : a + b = 4} is a relation on N, then R is
(a) reflexive (b) symmetric
(c) anti symmetric (d) transitive

Answer 1

Answer:

symmetric

Step-by-step explanation:

When a relation is equal to its inverse, then the relation is symmetric.

Answer 2

We need to recall the following definitions of relations.

For relation $R$ on a set $A$

• Reflexive: A relation is reflexive if $(a,a)\in R$ for every $a\in A$.

• Symmetric: A relation is symmetric if $(a,b)\in R$ then $(b,a) \in R$.

• Transitive: A relation is symmetric if $(a,b)\in R$ and $(b,c) \in R$ then $(a,c)\in R$.

• Anti-symmetric: the relation $R$ is antisymmetric if either $(a,b)\notin R$ or $(b,a)\notin R$ whenever $a\neq b$.

Given:

$R = {(a, b) : a + b = 4}$ is a relation on $N$

We get,

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

Since $(1,3)\in R$ so this relation is not reflexive.

The relation is not transitive as $(1,3,)\in R$, $(3,1)\in R$ but $(1,1)\notin R$.

The relation is not anti-symmetric as $(1,3,)\in R$, $(3,1)\in R$ but $1\neq 3$

The relation is symmetric as $(1,3,)\in R$, $(3,1)\in R$  and $(2,2)\in R$.

So the relation is only symmetric.

Hence, the correct option is b) symmetric.