Determine whether the relation R defined below is reflexive, irreflexive, symmetric, asymmetric, antisymmetric, or transitive. For each property, either explain why R has that property or give an example showing why it does not.

a) Let A = {1, 2, 3, 4} and let R = { (2, 3) }

b) Let A = {1, 2, 3, 4} and let R = { (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 4), (3, 1), (3, 3), (4, 1), (4, 4) }.

See below

Step-by-step explanation:

Remember some definitions about binary relations. If R⊆S*S then

R is reflexive if (a, a) ∈R for all a∈S R is irreflexive if (a, a) ∉R for all a∈S R is symmetric if (a, b) ∈R implies (b, a) ∈R for all a, b∈S R is asymmetric if (a, b) ∈R implies (b, a) ∉R for all a, b∈S R is antisymmetric if (a, b) ∈R and (b, a) ∈R imply that a=b, for all a, b∈S R is transitive if (a, b) ∈R and (b, c) ∈R imply (a, c) ∈R for all a, b, c∈S

a) R is not reflexive since (1,1) ∉R.

R is irreflexive, since (a, a) ∉R for all a=1,2,3,4

R is asymmetric: (2,3) ∈R and (3,2) ∉R (thus R is not symmetric).

R is antisymmetric, there are no cases to check. R is transitive, there are no cases to check.

b) R is reflexive, checking case by case, (a, a) ∈R for all a=1,2,3,4. Hence R is not irreflexive.

R is not asymmetric: (1,2) ∈R but (2,1) ∈R. R is not symmetric, since (4,1) ∈R but (1,4) ∉R

R is not antisymmetric: (1,2) ∈R and (2,1) ∈R but 1≠2.

R is not transitive: (1,2) ∈R and (2,4) ∈R but (1,4) ∉R.