Consider the binary operations`*: RxxR->R` and `o: RxxR->R` defined as `a*b=|a-b|` and `aob=a` for all `a ,\ b in Rdot` Show that `*` is commutative but not associative, `o` is associative but not commutative. Further, show that * is distributive over `o` . Dose `o` distribute over * ? Justify your answer.

Explanation

Let check commutative for *

* is commutative if

`a`*`b=b`*`a`

`a`*`b=∣a−b∣`;`b`*`a=∣b−a∣=∣a−b∣`

Since,

`a`*`b=b`*`a` for all `a,b in R`

Hence * is commutative.

Now check associative for *

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