Show that * : `R xx R to R` defined by: `a * b =a-5b` is not abelian.

Show that *: R xxR ->R defined by a*b = a +2b is not commutative.

Prove that **: R xx R to R defined as a ** b =a+2ab is not commutative .

Prove that **: R xx R to R defined as a ** b =a+2ab is not associative

Show that *: Rxx R ->R given by a*b = a +2b is not associative.

Consider the binary operation * : R xx R rarr R and o : R xx R rarr R defined as a * b = |a-b| and a o b = a, AA a, b in R . Is o distributive over * ? Justify your answer.

Consider the binary operations ** \: R\ xx\ R -> R and o\ : R\ xx\ R -> R defined as a **\ b=|a-b| and aob=a for all a,\ b\ in R . Show that ** is commutative but not associative, o is associative but not commutative.

Show that the relation R defined by R={(a , b):a-b is divisible by 3; a , bZ} is an equivalence relation.

Show that the relation R defined by R={(a , b):a-b is divisible by 3; a , bZ} is an equivalence relation.

Show that the relation R defined by R={(a , b):a-b is divisible by 3; a , bZ} is an equivalence relation.

Show that the relation R on R defined as R={(a ,\ b): alt=b} , is reflexive and transitive but not symmetric.