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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
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
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