{F} Construct a relation on the set {a, b, c, d} that is a. reflexive, symmetric, but not transitive. b. irreflexive, symmetric, and transitive. c. irreflexive, antisymmetric, and not transitive. d. reflexive, neither symmetric nor antisymmetric, and transitive. e. neither reflexive, irreflexive, symmetric, antisymmetric, nor transitive.

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