Prove that the operation * on the set `M={[(a,0),( 0,b)]: a ,\ b in R-{0}}` defined by `A`*`B=A B` is a binary operation.

Explanation

We have given that * is an operation that is valid on the set `M={[(a,0),( 0,b)]: a ,\ b in R-{0}}`and it is defined as given ` A`*`B = A B`

According to the problem it is given that on applying the operation * for two given numbers in the set ‘M’ it gives a number in the set ‘M’ as a result of the operation.

`implies A`*`BinM ...... (1)`

Let us take `AB = [(a,0),( 0,b)]xx[(c,0),( 0,d)]`

`⇒ AB=[(ac,0),( 0,bd)]`

Since `ainR` and `cinR` then `acinR` And also `binR` and `dinR` then `bdinR`

`implies ABinR`

Therefore the operation * defines a binary operation on ‘M’.