Question 1 :
Let $$T$$ be the set of all triangles in the Euclidean plane, and let a relation $$R$$ on $$T$$ be defined as $$aRb$$, if $$a$$ is congruent to $$b$$ for all $$a,b\in T$$. Then, $$R$$ is

Question 3 :
On the set $$N$$ of all natural numbers define the relation $$R$$ by $$a R b$$ if and only if the G.C.D. of $$a$$ and $$b$$ is $$2$$. Then $$R$$ is:

Question 4 :
Let $$f: N\rightarrow R$$ such that $$f(x)=\dfrac{2x-1}{2}$$ and $$g: Q\rightarrow R$$such that $$g(x)=x+2$$ be two function. Then $$(gof)\left(\dfrac{3}{2}\right)$$ is equal to

Question 5 :
Find number of all such functions $$y = f(x)$$ which are one-one?

Question 6 :
If $$R$$ is an equivalence relation in a set $$A$$, then $$R^{-1}$$ is

Question 8 :
Let Z be the set of all integers and let R be a relation on Z defined by $$a$$ R $$b\Leftrightarrow (a-b)$$ is divisible by $$3$$. Then, R is?

Question 9 :
 $$f : R \rightarrow R , f ( x ) = e ^ { | x | } - e ^ { - x }$$  is many-one into function.

Question 10 :
Let $$Z$$ be the set of integers and $$aRb$$, where $$a, b\epsilon Z$$ if an only if $$(a - b)$$ is divisible by $$5$$.<br>Consider the following statements:<br>$$1.$$ The relation $$R$$ partitions $$Z$$ into five equivalent classes.<br>$$2.$$ Any two equivalent classes are either equal or disjoint.<br>Which of the above statements is/are correct?

Question 11 :
In order that a relation $$R$$ defined in a non-empty set $$A$$ is an equivalence relation, it is sufficient that $$R$$

Question 12 :
If $$\displaystyle A= \left \{ x:-1\leq x\leq 1 \right \}=B.$$&#160;Discuss the following function w.r.t one-one onto bijective and write their characteristics.$$\displaystyle f\left ( x \right )=\frac{x}{2}$$

Question 13 :
The minimum number of elements that must be added to the relation $$R=\left \{ (1, 2),(2, 3) \right \}$$ on the set of natural numbers, so that it is an equivalence is:

Question 14 :
If A ={1, 3, 5, 7} and B = {1, 2, 3, 4, 5, 6, 7, 8}, then the number of one-to-one functions from A into B is

Question 15 :
Let $$R = \{(2,3),(3, 4)\}$$ be relation defined on the set of natural numbers. The minimum number of ordered pairs required to be added in $$R$$ so that enlarged relation becomes an equivalence relation is

Question 17 :
Which of the following functions from $$Z$$ to itself are bijections?

Question 18 :
The function $$f: [0, 3]$$ $$\rightarrow$$ $$[1, 29]$$, defined by $$f(x) = 2x^3-15x^2 + 36x+ 1$$, is<br>

Question 19 :
Let $$\displaystyle f\left ( x \right )=\frac{ax^{2}+2x+1}{2x^{2}-2x+1}$$, the value of $$a$$ for which $$\displaystyle f:R\rightarrow \left [ -1,2 \right ]$$ is onto , is<br>

Question 20 :
Let f: $$X\rightarrow Y$$ be a function defined by $$f(x)=a \sin \left (x+\dfrac {\pi}{4}\right )+b \cos x+c$$. If f is both one-one and onto, then find the sets $$X$$ and $$Y$$