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| Math Central | Quandaries & Queries |
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Question from jim: prove or disprove: Every binary operation on a set consisting of a single element is both commutative and associative. Every commutative binary operation on a set having just two elements is associative. |
Hi Jim,
A binary operation × on a set A is commutative if for each element a and b in A, a × b = b × a.
If there is only one element in A then a = b. Is the condition above true if a = b?
For the associate property write the definition and then replace the name of each element by the same letter. Is the condition true?
For your last question there are only a few possibilities. If the two elements are a and b then
a × b = b × a = a or b
a × a = a or b
b × b = a or b
Try the various combinations and see if each satisfies the associate property.
Penny
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