General Doubt in relation n absolute value

2)

relation means a set of all possible (a,b) where aεA and bεB

a can be chosen in m ways.

b can be chosen in n ways for each a...

thus total of mn ordered pairs are possible..[1]

Now when we are building a function we can either take in one of the ordered pair or reject it to get lost! [3]

So each ordered pair can be treated in 2 ways...

thus net number of possible collection=2^mn

but this includes when none is selected...i.e. a null set and it has to be ignored as it is not a relation..

thus total number of relations=2^mn-1

3)

for this sum i ll upload the figure of one section of the entire setup .. this section contains the axis of the cylinder in consideration! [1]
the section:

R=radius of the sphere
r=radius of cylinder
x=height of cylinder

from the right angled triangle,
r^2+\left(\frac{x}{2} \right)^2=R^2\\ or,\;r^2=\frac{4R^2-x^2}{4}

We know,

V(x)= \pi r^2x = \pi\left(\frac{4R^2-x^2}{4} \right)x\\ or,\;\frac{V(x)}{x} = \frac{\pi}{4}\left(4R^2-x^2 \right)

Now u decide from the options! [1]

Nice one Ashish! [1]

1) an unordered pair or pair set is a set of the form {a, b}, i.e. a set having two elements a and b with no particular relation between them. In contrast, an ordered pair (a, b) has a as its first element and b as its second element.

2) disjoint means the intersection of that 2 set is a null set!

3) null set is sub set of every set! [1]

So these unordered pairs have 2 elements each of which is another set..

let the element sets be A and B

{n(A),n(B)} can be {0,1}, {0,2}, {0,3}, {1,1}, {1,2}

and none else (why?)

{n(A),n(B)} = {0,1} can be of 3 ways..
{n(A),n(B)} = {0,2) can be of 3 ways..
{n(A),n(B)} = {0,3} can be of 1 way..
{n(A),n(B)} = {1,1} can be of 3 ways..
{n(A),n(B)} = {1,2} can be of 3 ways..

thus net 13 ways..

Did I miss something out???

sorry i took S={1,2,3}
anyway the essence of the solution remains the very same! [1]

So, the post thrives..

try solving out the original sum urself! [1]