Question

Consider the binary operations *: R × R → R and o: R × R → R defined as a∗b=|a−b| and a o b = a, ∀ a, b ∈ R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that ∀ a, b, c ∈ R, a*(b o c) = (a * b) o (a * c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.

Answer 1

Hence, the operation * is commutative. It can be observed that

∴ a o b) o c = a o (b o c), where a, b, c ∈ R 
Hence, the operation o is associative.

Hence, the operation o does not distribute over *.