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

Show that *: R xx R to R defined by a*b = 3a + 5b is non commutative and non associative:

Show that **: R x R to R defined by a **b=a + 2b is not commutative.

Show that ** R xxR to R given by a ** b to a + 2b is not associative.

Show that **: R xx R to R given by (a,b) to a + 4b ^(2) is a binary operation.

Show that +: R xx R to R and x : R xx R to R are communtative binary opertions , but - : R xx R to R and div : R _(**) xx R _(**) to R_(**) are not commutative.

Show that the vv:R xx R given by (a,b) to max {a,b} and the ^^ : R xx R to R given by (a,b) to min {a,b} are binary operations.

Let R be a relation from Q to Q defined by R={(a,b): a,b in Q and a-b in Z} . Show that (i) (a,a) in R" for all " a in Q

Let S be the set of real numbers. For a, b in S , relation R is defined by aRb iff |a-b|lt 1 then R is