{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.

 a) reflexive, symmetric, but not transitive.

consider a relation {a, b, c ,d} such that

R1 = { (a, a), (b, b), (c, c), (d, d), (a, b), (b, a), (b, c), (c, b) }

we may see that for all x$\isin$ {a, b, c, d}, (x, x)$\isin$ R1, thus R1 is reflexive.

Also, (b, a)$\isin$R1 whenever (a, b)$\isin$ R1 and (c, b)$\isin$ R1 whenever (b, c)$\isin$ R1, thus we may conclude that R1 is symmetric.

Also if (a, b)$\isin$ R1 and (b, c)$\isin$ R1 but (a, c) $\notin$ R1. Hence we conclude that R1 is not transitive.

b) irreflexive, symmetric, and transitive.

let R2=$\varnothing$ ,Consider a, b $\isin$ {a, b, c, d}, if bRa then aRb, hence R is symmetric.

Now let a, b, c $\isin$ {a, b, c, d}, if cRa then aRb or bRc, hence R is transitive.

Now taking any element a$\isin$ S and observe that aRa. Then R is not reflexive, hence it is irreflexive.

c). irreflexive, antisymmetric, and not transitive.

Consider a relation {a, b, c, d} such that R3 = { (a, b), (b, c)}

we may see that (a, a)$\isin$ R3 so R3 is not reflexive and is thus irreflexive.

Also if a=b whenever (a, b)$\isin$ R3 and (b, a)$\isin$ R3, but for this case (b, a)$\notin$ R3 , So it is antisymmetric.

Also if (a, b)$\isin$ R1 and (b, c)$\isin$ R1 but (a, c)$\notin$ R1. Hence R3 is not transitive.

d.) reflexive, neither symmetric nor antisymmetric, and transitive.

consider a relation {a, b, c, d} such that R4= { (a, a), (b, b), (c, c), (d, d), (a, b), (b, a), (c, a), (b, c) }

we can see that for x$\isin$ {a, b, c, d}, (x, x)$\isin$ R4, thus R4 is reflexive.

Also (b, c)$\isin$ R4 but (c, b)$\notin$ R4, thus we say that R4 is not symmetric. Now, (b, a)$\isin$ R4 whenever (a, b)$\isin$ R4 but a is not equal to b, thus R4 is not antisymmetric.

Also, if (a, b)$\isin$ R4 and (b, c)$\isin$ R4 then (a, c)$\isin$ R4. Hence R4 is transitive

e). neither reflexive, irreflexive, symmetric, antisymmetric, nor transitive.

consider a relation {a, b, c, d) such that R5 = { (a, b) , (b, a), (c, c) , (a, c) }

we can see that( a, a)$\notin$ R5 so R5 is not reflexive and also we can see that (c, c)$\isin$R5 means it is not reflexive.

Also, (a, c) $\isin$ R5 but (c, a)$\notin$ R5 thus we can say R5 is not symmetric. Now (b, a)$\isin$ R5 whenever (a ,b) $\isin$ R5 but a is not equal to b, thus R5 is not antisymmetric.

Also, if (b, a)$\isin$ R5 and (a, c)$\isin$ R5 then (b, c)$\notin$ R5. Hence R5 is not transitive.