Question

Which of the following is not a binary operation on the indicated set?

Options

• On Z⁺, * defined by a * b = a – b

• On Z⁺, * defined by a * b = ab

• On R⁺, * defined by a * b = ab²

• None of the above

Answer 1

On Z⁺, * defined by a * b = a – b

Explanation:

On Z⁺, * defined by a * b = a – b

It is not a binary operation as the image of (1, 2)under * is 1 * 2 = 1 – 2 = – 1 ∉ Z⁺.

 On Z⁺, * is defined by a * b = ab

It is seen that for each a, b ∈ Z⁺, there is a unique element ab in Z⁺.

This means that * carries each pair (a, b) to a unique element a * b = ab in Z⁺.

Therefore, * is a binary operation.

On R, *is defined by a * b = ab².

It is seen that for each a, b ∈ R, there is a unique element ab² in R.

This means that * carries each pair (a, b) to a unique element a * b = ab² in R.

Therefore, * is binary operation.