Question

1.
Show that the operation * given by X*y=x+y-xy is a binary operation on Z, Q and R but
not on N.
​

Answer 1

A binary operation on a set must be closed under the operation (that is that the resulting number must also be in the same set).

The answer to this exercise depends on the properties you have previously learned about the mentioned sets as well as addition, multiplication and subtraction. This answer will assume that you can use the closure of addition, multiplication and subtraction

*  is clearly a binary operation on $\mathbb{Z}$ as addition, multiplication and subtraction of integers always results in other integers (closure over $\mathbb{Z}$.)

Similarly addition, multiplication and subtraction of rational and real numbers always results in other rational and real numbers respectively.

However, the natural numbers $\mathbb N$, are closed under addition and multiplication, but not subtraction. Hence, the operation x * y = x + y - xy can result in negative numbers, which are not elements of the natural numbers. One such counter example is $2*3=2+3 - 2\times 3 = 5 - 6 = -1 \notin \mathbb N$

If you're not allowed to use the closure of the basic arithemtic operations you can try to show by contradiction that * must be closed over the different sets.