Given an example of a relation. Which is

Solution:

(i) Let A = {5, 6, 7}.

Define a relation R on A as R = {(5, 6), (6, 5)}.

Relation R is not reflexive as (5, 5), (6, 6), (7, 7) ∉ R.

Now, as $(5,6) \in R$ and also $(6,5) \in R, R$ is symmetric.

$\Rightarrow(5,6),(6,5) \in R$, but $(5,5) \notin R$

∴R is not transitive.

Hence, relation R is symmetric but not reflexive or transitive.

(ii)Consider a relation R in R defined as:

$R=\{(a, b): a

For any $a \in R$, we have $(a, a) \notin R$ since $a$ cannot be strictly less than $a$ itself. In fact, $a=a$.

∴ R is not reflexive.

Now,

$(1,2) \in R($ as $1<2)$

∴ R is not symmetric.

Now, let $(a, b),(b, c) \in R$.

$\Rightarrow a

$\Rightarrow a

$\Rightarrow(a, c) \in R$

∴R is transitive.

Hence, relation R is transitive but not reflexive and symmetric.

(iii)Let A = {4, 6, 8}.

Define a relation R on A as:

$A=\{(4,4),(6,6),(8,8),(4,6),(6,4),(6,8),(8,6)\}$

Relation $R$ is reflexive since for every $a \in A,(a, a) \in R$ i.e., $(4,4),(6,6),(8,8)\} \in R$.

Relation $R$ is symmetric since $(a, b) \in R \Rightarrow(b, a) \in R$ for all $a, b \in R$.

Relation $R$ is not transitive since $(4,6),(6,8) \in R$, but $(4,8) \notin R$.

Hence, relation R is reflexive and symmetric but not transitive.

(iv) Define a relation R in R as:

$\left.R=\{a, b): a^{3} \geq b^{3}\right\}$

Clearly $(a, a) \in R$ as $a^{3}=a^{3}$

∴R is reflexive.

Now,

$(2,1) \in R\left(\operatorname{as} 2^{3} \geq 1^{3}\right)$

∴ R is not symmetric.

Now,

Let $(a, b),(b, c) \in \mathrm{R}$.

$\Rightarrow a^{3} \geq b^{3}$ and $b^{3} \geq c^{3}$

$\Rightarrow a^{3} \geq c^{3}$

$\Rightarrow(a, c) \in R$

∴R is transitive.

Hence, relation R is reflexive and transitive but not symmetric.

(v) Let $A=\{-5,-6\}$.

Define a relation R on A as:

$R=\{(-5,-6),(-6,-5),(-5,-5)\}$

Relation $R$ is not reflexive as $(-6,-6) \notin R$.

Relation $R$ is symmetric as $(-5,-6) \in R$ and $(-6,-5\} \in R$.

It is seen that $(-5,-6),(-6,-5) \in \mathrm{R}$. Also, $(-5,-5) \in \mathrm{R}$.

∴The relation R is transitive.

Hence, relation R is symmetric and transitive but not reflexive.