On the set of integers Z, define f : Z $\to$ Z as f(n) = $$\begin{gathered} \left\{\frac{\mathrm{n}}{2}, \mathrm{n} \mathrm{is} \mathrm{even} \\ 0, \mathrm{n} \mathrm{is} \mathrm{odd}\right\} \end{gathered}$$, then 'f' is

• injective but not surjective

• neither injective nor surjective

• surjective but not injective

• bijective

The inverse of 2010 in the group Q^* of all positive rational under the binary operation * defined by a * b = $\frac{\mathrm{ab}}{2010}, \forall \mathrm{a}, \mathrm{b} \in {\mathrm{Q}}^{+}$ is

• 2009

• 2011

• 1

• 2010

Define a relation R on A = {1, 2, 3, 4} as xRy if x divides y. R is

• reflexive and transitive

• reflexive and symmetric

• symmetric and transitive

• equivalence

On the set of all non-zero reals, an operation * is defined as a * b = $\frac{3\mathrm{ab}}{2}$. In this group, a solution of (2 * x) * 3^-1 = 4^-1 is

• 6

• 1

• 1/6

• 3/2

If A and B have n elements in common, then the numberofelements common to A x B and B x A is

• n

• 2n

• n²

• 0

Which of the following is false ?

• (N, *) is a group
  
  

• (N, +) is a semi-group

• (Z, +) is a group

• Set of even integers is a group under usual addition

Let S be the set of all real numbers. A relation R has been defined on S by aRb $\iff \left|\mathrm{a} - \mathrm{b}\right| \le 1$, then R is

• symmetric and transitive but not reflexive

• reflexive and transitive but not symmetrIc

• reflexive and symmetric but not transitive

• an equivalence relation

For any two real numbers, an operation * defined by a * b  = 1 + ab is

• neither commutative nor associative

• commutative but not associative

• both commutative and associative

• associative but not commutative

Let f : N $\to$ N defined by f(n) = $$\begin{gathered} \left\{\frac{\mathrm{n} +1}{2}, \mathrm{if} \mathrm{n} \mathrm{is} \mathrm{odd} \\ \frac{\mathrm{n}}{2}, \mathrm{if} \mathrm{n} \mathrm{is} \mathrm{even}\right\} \end{gathered}$$, then f is

• onto but not one-one

• one-one and onto

• neither one-one nor onto

• one-one but not onto

Suppose f(x) = (x + 1)² for x $\ge$ - 1. If g(x) is a function whose graph is the reflection of the graph of f(x) in the line y = x, then g(x) is equal to

• $\frac{1}{{\left(\mathrm{x} + 1\right)}^2}\mathrm{x} > - 1$

• $- \sqrt{\mathrm{x}} - 1$

• $\sqrt{\mathrm{x}} + 1$

• $\sqrt{\mathrm{x}} - 1$