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 Determine whether or not the operation * defined on Z+, defined by a*b = a-b gives a binary operation. If the event * is not a binary operation, give justification of this.


Answer
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Hint: A binary operation * on a set A is a function from A×AA. Hence * should be defined on every element of A×A and the set A should be closed under * for * to be a binary operation on A.

Complete step-by-step answer:

The set Z+ is defined as Z+={n:nZ,n0}
The operation * defined as a*b = a-b over Z+ does not form a binary operation because Z+ is not closed under Z+.
Consider the case when a = 2 and b = 3.
We have aZ+ and bZ+.
So (a,b)Z+×Z+
But a*b = a - b = 2 - 3 = -1
Since 1Z+, Z+ is not closed under *.
In other words there exists an ordered pair (a,b)Z+×Z+ such that (a,b) is unmapped by *.
So, * is not a function from Z+×Z+Z+ and hence * is not a binary operation

Note:
[1] In order to prove that a statement is incorrect it is sufficient to come up with a counter example. In the question we proved that * is not a binary operation by proving that the claim of * being a binary operation will be incorrect as we have come up with a counter example of a = 2 and b = 3.
[2] A binary operation * is said to be commutative if a*b = b*a
[3] A binary operation * is said to be associative if a*(b*c) = (a*b)*c
[4] Addition forms a binary operation over the  set of Natural numbers.
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