State whether the following statements are true or false. Justify.
(i). For an arbitrary binary operation * on a set N, a * a = a for all a ∈ N.
(ii). If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a

Solution:

The commutative property deals with the arithmetic operations of addition and multiplication. It means that changing the order or position of numbers while adding or multiplying them does not change the end result

(i). Define operation * on a set N as a * a

= a for all a ∈ N

In particular, for a = 3,

3 * 3 = 9 ≠ 3

Therefore, statement (i) is false.

(ii). R.H.S. = (c * b) * a

= (b * c) * a [* is commutative]

= a * (b * c) [Again, as * is commutative]

= L.H.S.

⇒ a * (b * c) = (c * b)* a

Therefore, statement (ii) is true

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

State whether the following statements are true or false. Justify. (i). For an arbitrary binary operation * on a set N, a * a = a for all a ∈ N. (ii). If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a

Summary:

(i). For an arbitrary binary operation * on a set N, a * a = a for all a ∈ N is not true. (ii). If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a is true