Question 1166868: Determine whether the definition of ⋆ does give a binary operation on the set. In case that ⋆ is not a binary operation, state the property it fails to satisfy and give a counterexample.
a). Let S = ℕ, define ⋆ by a ⋆ b=c, where in c is the largest integer less than the product of a and b.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
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Response revised/corrected....
The two inputs are integers; and the output is always an integer.
That makes * a binary operation on the set of integers.
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Revision...
I wrongly took the given set to be the set of positive integers; in standard nomenclature it is the set of natural numbers.
Even if the set were the set of positive integers, there would be a single counterexample making * NOT a binary operation on the set:
1*1 = 1-1 = 0
But 0 is not a positive integer.
With the set actually being the natural numbers, there are an infinite number of counterexamples. If either a or b is 0, then a*b = 0-1 = -1; and -1 is not a natural number.
ANSWER: a*b does NOT define a binary operation on the given set.
Answer by ikleyn(52784)  (Show Source):
You can put this solution on YOUR website! .
The post / (the condition) is FATALLY INCOMPLETE, since it does not determine what the set N is.
The standard designation for the set of integer numbers is Z;
so, based on the post, the set of numbers in this problem is not Z;
then WHAT it is ?
It is a typical case at this forum, when the person who creates / (composes) the problem,
does not know the subject and does not know WHAT he (or she) writes about and how to write it correctly.
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From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Binary_operation#:~:text=In%20mathematics%2C%20a%20binary%20operation,operands)%20to%20produce%20another%20element.&text=Examples%20include%20the%20familiar%20arithmetic%20operations%20of%20addition%2C%20subtraction%2C%20multiplication.
In mathematics, a binary operation or dyadic operation is a calculation that combines two elements (called operands)
to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, a binary operation on a set is an operation whose two domains and the codomain are the same set.
Examples include the familiar arithmetic operations of addition, subtraction, multiplication.
Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication and conjugation in groups.
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