For the set $$N$$ of natural numbers and a binary operation $$f:N \times N \to N$$, an element $$z \in N$$ is called an identity for $$f$$ if $$f\left( {a,z} \right) = a = f\left( {z,a} \right)$$ for all $$a \in N$$. Which of the following binary operations have an identify?
$${\rm I}$$) $$\,\,\,\,\,\,f\left( {x,y} \right) = x + y - 3$$
$${\rm I}{\rm I}$$ $$\,\,\,\,\,\,f\left( {x,y} \right) = {\mkern 1mu} \max \left( {x,y} \right)$$
$${\rm I}{\rm I}{\rm I}$$$$\,\,\,\,\,f\left( {x,y} \right) = \,{x^y}$$

A relation $$R$$ is defined on ordered pairs of integers as follows: $$\left( {x,y} \right)R\left( {u,v} \right)\,if\,x < u$$ and $$y > v$$. Then $$R$$ is

The set $$\left\{ {1,\,\,2,\,\,3,\,\,5,\,\,7,\,\,8,\,\,9} \right\}$$ under multiplication modulo 10 is not a group. Given below are four plausible reasons.

Which one of them is false?

Let $$X,. Y, Z$$ be sets of sizes $$x, y$$ and $$z$$ respectively. Let $$W = X x Y$$ and $$E$$ be the set of all subjects of $$W$$. The number of functions from $$Z$$ to $$E$$ is