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.

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 is a multiple of 4, then 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.