State whether the following statements are true or false. Justify.

Question:

State whether the following statements are true or false. Justify.

(i) For an arbitrary binary operation * on a set N N.

(ii) If * is a commutative binary operation on N, then * (c) = (b) * a

Solution:

(i) Define an operation * on N as:

* b = a + b a, b ∈ N

Then, in particular, for b = a = 3, we have:

3 * 3 = 3 + 3 = 6 ≠ 3

Therefore, statement (i) is false.

(ii) R.H.S. = (* b) * a

= (* c) * [* is commutative]

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

= L.H.S.

∴ a * (* c) = (* b) * a

 

Therefore, statement (ii) is true.

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