Question

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) }.

Answer 1

Answer:

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.