Question

{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.

Answer 1

 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)$\isin$ R_1, 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 bR_a then aR_b, hence R is symmetric.

Now let a, b, c $\isin$ {a, b, c, d}, if cR_a then aR_b or bR_c, hence R is transitive.

Now taking any element a$\isin$ S and observe that aR_a. 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.