
Determine whether or not the operation * defined on , 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 . Hence * should be defined on every element of and the set A should be closed under * for * to be a binary operation on A.
Complete step-by-step answer:
Note:
Complete step-by-step answer:
The set is defined as
The operation * defined as a*b = a-b over does not form a binary operation because is not closed under .
Consider the case when a = 2 and b = 3.
We have and .
So
But a*b = a - b = 2 - 3 = -1
Since , is not closed under *.
In other words there exists an ordered pair such that is unmapped by *.
So, * is not a function from 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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