Question

In: Advanced Math

For each of the following determine whether ∗ is a binary operation on R. If so,...

For each of the following determine whether ∗ is a binary operation on R. If so, determine whether or not ∗ is associative, commutative, has an identity element, and has inverse elements.

(a) a ∗ b = (ab) / (a+b+1)

(b) a ∗ b = a + b + k where k ∈ Z

Solutions

Expert Solution

a) No it is not a binary operation on R. This is because it not well defined for for which the output is not well defined

b) For a fixed value of k, this is a binary operation as is also a real number

It is associative as

It is commutative as

The identity element is as

Inverse of is since

4) For the given subset of reals, is well defined so that it is a binary operation

It's not commutative as

It is not associative as and which are not equal

And which is not unique

Thus a unique identity does not exist

As it has no inverse, an identity cannot exist

Hope this helped. Please do rate positively. Thanks and have a good day.

Colby Messinger answered 1 year ago

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