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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.

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konrad509

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 [tex]a+b[/tex] is a multiple of 4, then [tex]b+a[/tex] 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.

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