Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation, give justification for this.
(i). On Z⁺, define * by a * b = a - b (ii). On Z⁺, define * by a * b = ab (iii). On R , define * by a * b = ab² (iv). On Z⁺, define * by a * b = |a - b| (v). On Z⁺, define * by a * b = a

Solution:

i. On Z⁺, define * 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⁺.

Therefore, * is not a binary operation.

ii. On Z⁺, define * 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.

iii. On R , define * 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 a binary operation.

iv. On Z⁺, define * by a * b = |a - b|

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

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

Therefore, * is a binary operation.

v. On Z⁺, define * by a * b = a

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

Therefore, * is a binary operation

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.4 Question 1

Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation, give justification for this.
(i). On Z⁺, define * by a * b = a - b (ii). On Z⁺, define * by a * b = ab (iii). On R , define * by a * b = ab² (iv). On Z⁺, define * by a * b = |a - b| (v). On Z⁺, define * by a * b = a

Summary:

i) * is not a binary operation ii) * is a binary operation iii) * is a binary operation iv) * is a binary operation v) * is a binary operation