Relations Test ...

Let R be a relation on the set of integers given by $$ aRb \leftrightarrow a = 2^k .b $$ for some integer $$k.$$ Then $$R$$ is 

Total number of equivalence relations defined in the set $$S=\left\{a, b, c\right\}$$ is?

Let $$A={1,2,3}$$. Then number of equivalence relations containing $$(1,2)$$ is

If R is a reflexive relation on a finite set A having n-elements, and there are m ordered pairs in R. then?

Let R be a relation from $$ A=\{1,2,3,4\}$$ to $$B=\{1,3,5\}$$such that R=[(a,b):a<b where a\in{A} and b\in{B}].What is $$RoR^{-1} $$ equal to

Let $$R$$ a relation on the set $$N$$ be defined by $$\left\{ \left( x,y \right) |x,y\in N,2x+y=41 \right\}$$. Then $$R$$ is 

$$f:\left( { - \infty ,\infty } \right) \to \left( {0,1} \right]$$ defined by $$f\left( x \right) = \frac{1}{{{x^2} + 1}}$$ is 

Let $$X=\left\{ x \epsilon {R};\cos(\sin\ x)=\sin(\cos\ x)\right\}$$. The number of elements in $$X$$ is 

If the relation $$R:A \to B$$ where $$A = \left\{ {1,2,3,4} \right\},B = \left\{ {1,3,5} \right\}$$ is defined by $$R = \left\{ {\left( {x,y} \right):x < y,x \in A,y \in B} \right\}$$ then $$Ro{R^{ - 1}} = $$ 

For real number $$x$$ and $$y$$, define a relationship $$R$$ {;$$x\ Ry$$ if and only if $$x-y+\sqrt{2}$$ is an irrational number }. Then the relation $$R$$ is