Math CentralQuandaries & Queries


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.


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