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## Here's the Solution to this Question

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)$\isin$ Rand (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 Ris not antisymmetric.

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