Embibe Experts Solutions for Chapter: Relations and Functions, Exercise 1: Exercise

Determine whether the given definition of $*$ gives a binary operation on the corresponding given set: On $Z^{+}$, define $* \text{by} a*b=c$, where $c$ is the largest integer less than the product of $a \text{and} b$.

State whether the following graph is a one-to-one function or not, giving a reason.

State whether the following graph is a one-to-one function or not, giving a reason.

State whether the following graph is a one-to-one function or not, giving a reason.

Give an example of a relation which is reflexive, symmetric but not transitive.

Show that the function $f:R\to R$ defined by $f\left(x\right)=\frac{x}{x^2+1},\forall x\in \mathrm{ℝ}$ is neither one-one nor onto. Also, if $g:R\to R$ is defined as $g\left(x\right)=2 x-1$, find $fog\left(\mathrm{x}\right)$.

Give an example of a relation which is reflexive, transitive but not symmetric

Let S be a set with 1 element.Then the number of different binary operation that can be defined on S in  _____