Aptitude | GATE IT 2006 | Question 2

For the set N of natural numbers and a binary operation f : N x N → N, an element z ∊ N is called an identity for f, if f (a, z) = a = f(z, a), for all a ∊ N. Which of the following binary operations have an identity?
  1. f (x, y) = x + y – 3
  2. f (x, y) = max(x, y)
  3. f (x, y) = xy

 
(A) I and II only
(B) II and III only
(C) I and III only
(D) None of these


Answer: (A)

Explanation: I f(x,y) = x+y-3 = x= y+x-3  =>  y=3 Here identity elements is 3
II f(x,y) = max(x,y)=x=max(y,x)  => y=1 Here identity elements is 1
(III f(x,y) =x^y is not same as f(y,x) = y^x. So no identity element.

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