Set Theory Test...

If $$S_k=\begin{bmatrix} 1 & k \\ 0 & 1\end{bmatrix}$$, k$$\in N^+$$, where N is the set of all natural numbers, then $$(S_2)^n(S_k)^{-1}$$ for n$$\in$$N is?

All the permissible values of $$b$$, if $$a=0$$ and $${S}_{2}$$ is a subset of $$\left( 0,\pi  \right) $$

If B= { $$y_{1}, y_{2}, y_{3} $$} and A = {$$x_{1}, x_{2}, x_{3}, .......x_{8}$$} having 3 elements in set B and 8 elements in set A. Then if the number of ways that $$ f: A\rightarrow B$$ is onto such that exactly 4 elements of $$ x \epsilon A$$  are matched to $$ y \epsilon B$$ such that $$ f(x) = y_{3} $$ is $$(_{r}^{n}\textrm{C}).m$$, then the value of $$m + r - n$$ is

State which of the following is total number of reflexive relations form set $$A = \left \{a, b, c\right \}$$ to set $$B = \left \{d, e\right \}$$ is

Sets $$A$$ and $$B$$ have $$5$$ and $$6$$ elements respectively and $$\left( A\triangle B \right) =C$$ then the number of elements in set $$\left( A-\left( B\triangle C \right)  \right)$$ is 

Suppose $${ A }_{ 1 },{ A }_{ 2 },,{A }_{ 30 }$$ are thirty sets each having $$5$$ elements and $${ B }_{ 1 },{ B }_{ 2 },..,{B}_{ n }$$ are $$n$$ sets each with $$3$$ elements, let $$\displaystyle \bigcup _{ i=1 }^{ 30 }{ { A }_{ i } } =\bigcup _{ j=1 }^{ n }{ { B }_{ j } =S}$$ and each element of $$S$$ belongs to exactly $$10$$ of the $${A}_{i}s$$ and exactly $$9$$ of the $${B}_{j}s.$$ Then $$n$$ is equal to

Suppose $$A_1 , A_2,... A_{30}$$ are thirty sets each having 5 elements and $$B_1, B_2,..., B_n$$ are n sets each with 3 elements , let $$\underset{i = 1}{\overset{30}{\cup}} A_i = \underset{j = 1}{\overset{n}{\cup}} B_j = S$$ and each element of S belongs to exactly 10 of the $$A_i's$$ and exactly 9 of the $$B_j'S$$. then n is equal to

An investigator interviewed $$100$$ students to determine their preferences for the three drinks: milk (M), coffee(C) and tea (T). He reported the following: $$10$$ students had all the three drinks M, C, T; $$20$$ had M and C only; $$30$$ had C and T; $$25$$ had M and T; $$12$$ had M only; $$5$$ had C only; $$8$$ had T only. Then how many did not take any of the three drinks is?

Suppose $${ A }_{ 1 },{ A }_{ 2 },....{ A }_{ 30 },$$ are thirty sets each with five elements and $${ B }_{ 1 },{ B }_{ 2 },....B_{ A },$$ are n sets each  with three elements such $$\overset { 30 }{ \underset { i-1 }{ U }  } \quad { A }_{ 1 }=\overset { n }{ \underset { j-1 }{ U }  } \quad =s$$ If each element of belongs to exactly ten of the $${ A }_{ 1 }$$ exactly 9 of the $${ A }_{ 1 }$$,then value of n is:

The value of $$\left( {A \cup B \cup C} \right) \cap \left( {A \cap {B^c} \cap {C^c}} \right) \cap {C^c}$$ is