Question

6) Let A = ℤ and R be the relation on A where a R b if and only if a + b is a multiple of 4. Determine whether R 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.

Answer 1

reflexive

NO - e.g. 1+1=2 which is not a multiple of 4

irreflexive

NO - e.g. 2+2=4 which is multiple of 4

symmetric

YES - If
`a+b` is a multiple of 4, then
`b+a` is all multiple of 4, because addition is commutative.

antysymmetric

NO, because it's symmetric.

transitive

NO - e.g. 1+3=4 , 3+5=8 , 1+5=6 and 6 is not a multiple of 4.