   Chapter 1.4, Problem 7TFE

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# True or FalseLabel each of the following statements as either true or false.Any binary operation defined on a set containing a single element is commutative and associative.

To determine

Whether the statement, “Any binary operation defined on a set containing a single element is commutative and associative” is true or false.

Explanation

Formula Used:

If x,yA, then a binary operation is said to be commutative if xy=yx is defined on set A.

Explanation:

Let a set A={x} with x as its arbitrary single element and a binary operation defined by xxx.

Then,

For commutative property,

{x}{x}={x}{x}={x}.

As, x is arbitrary element here so, for any singleton set commutative property is satisfied.

If x,y,zA, then a binary operation is said to be associative if (xy)z=x(yz) is defined on set A

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