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## Here's the Solution to this Question

Let us consider a set $\{a, b, c, d\}.$

Reflexivity: A relation R on set S is called reflexive if $(a, a) \in R$ for all $a \in S.$

Symmetry: A relation R on set S is called symmetric if $(b, a) \in R$ whenever $(a, b) \in R$ for all $a, b \in S.$

Anti symmetry: A relation R on set S is called anti symmetric if $(b, a) \in R$ and $(a, b) \in R$ than $a=b$ for all $a, b \in S.$

Transitivity: a relation R on set S is called transitive if $(a, b) \in R$ and $(b, c) \in R$ than $(a, c) \in R$ for all $a, b, c \in S.$

a) Consider a relation on $\{a, b, c, d\}$ such that

$R_{1}=\{(a, a),(b, b),(c, c),(d, d),(a, b),(b, a),(b, c),(c, b)\}$

Now we can see that for all $x \in\{a, b, c, d\},(x, x) \in R_{1}$ thus $R_{1}$ is reflexive.

Also $(b, a) \in R_{1}$ whenever $(a, b) \in R_{1}$ and $(c, b) \in R_{1}$ whenever $(b, c) \in R_{1}$ thus we can say $R_{1}$ is symmetric.

Again we can see that if $(a, b) \in R_{1}$ and $(b, c) \in R_{1}$ but $(a, c) \notin R_{1}$ Hence $R_{1}$ is not transitive.

b) Let $R_{2}=\varnothing$ . Consider on $a, b \in\{a, b, c, d\},$ if b R R a then a R b " hence the R is symmetric.

Let $a, b, c \in\{a, b, c, d\}$ , if c R a then a R b or b R R c " hence the R is transitivity.

Now take any element $a \in S$ and observe that $a \tilde{R} a$ . Thus R is not reflexive, hence it is irreflexive.

c) Consider a relation on $\{a, b, c, d\}$ such that

$R_{3}=\{(a, b),(b, c)\}$

We can see that $(a, a) \notin R_{3}$ so $R_{3}$ is not reflexive and also we can see that no element is related to each other means it is irreflexive.

Also we have if a=b whenever $(a, b) \in R_{3}$ and $(b, a) \in R_{3}$ but here $(b, a) \notin R_{3}$ so it is antisymmetric.

Again we can see that if $(a, b) \in R_{1}$ and $(b, c) \in R_{1}$ but $(a, c) \notin R_{1}$ Hence $R_{3}$ is not transitive.

d) Consider a relation on $\{a, b, c, d\}$ such that

$R_{4}=\{(a, a),(b, b),(c, c),(d, d),(a, b),(b, a),(c, a),(b, c)\}$

We can see that for all $x \in\{a, b, c, d\},(x, x) \in R_{4}$ thus $R_{4}$ is reflexive.

Also $(b, c) \in R_{4}$ but $(c, b) \notin R_{4}$ thus we can say $R_{4}$ is not symmetric. Now $(b, a) \in R_{4}$ whenever $(a, b) \in R_{4}$ but $a \neq b$ thus $R_{4}$ is not antisymmetric.

Again we can see that if $(a, b) \in R_{4}$ and $(b, c) \in R_{4}$ then $(a, c) \in R_{4}.$ Hence $R_{4}$ is transitive.

e) Consider a relation on $\{a, b, c, d\}$ such that

$R_{5}=\{(a, b),(b, a),(c, c),(a, c)\}$

We can see that $(a, a) \notin R_{5}$ so $R_{5}$ is not reflexive and also we can see that $(c, c) \in R_{5}$ means it is not irreflexive.

Also $(a, c) \in R_{5}$ but $(c, a) \notin R_{5}$ thus we can say $R_{5}$ is not symmetric. Now $(b, a) \in R_{5}$ whenever $(a, b) \in R_{5}$ but $a \neq b$ thus $R_{5}$ is not antisymmetric.

Again we can see that if $(b, a) \in R_{5}$ and $(a, c) \in R_{5}$ then $(b, c) \notin R_{5}.$ Hence $R_{5}$ is transitive