Cardinality of Sets

Q.

Draw appropriate Venn diagrams for each of the following :

(i) $(A\cup B{)}^{\prime }$ (ii) $A^{\prime }\cap B^{\prime }$

(iii) $(A\cap B{)}^{\prime }$ (iv) $A^{\prime }\cup B^{\prime }$

Q. 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 ${\bigcup }_{i=1}^{30}{A}_i={\bigcup }_{i=1}^n{B}_j=S$. Each element of S belongs to exactly 10 of the $A_{i}s$ and exactly 9 of the $B_{i}s$. Then n is equal to

1. 15
2. 50
3. 3
4. 45

Q.

Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1) are in $A\times B,$ find A and B, where x, y and z are distinct elements.

Q. 32. find the number of garlands that can be formed using 3 flowers of 1 kind and 12 flowers of 2nd kind is a two digit number AB.Find A and B.

Q. The value of the expression ${}^{10}{C}_0{10}^9{-}^{10}{C}_1{9}^9{+}^{10}{C}_2{8}^9.......{-}^{10}{C}_9$ is

1. 
2. 
3. 
4. 

Q.

Sonia has 10 balloons out of which 5 are red, 2 white, 2 blue and 1pink, which she wants to use for the decoration. Her favourite pink colour balloon should be filled with toffees and should be put at the centre of the room above the cake table and remaining 9 at the wall behind the cake table. How many ways she can arrange the balloons?

1. 956

2. 7560

3. 9!

4. 756

Q. For subsets A and B of a set X, define the set A\ast B as A\ast B=(A∩ B)∪((X-A)∩(X-B)). Then only one of the following statements is true, which one is it? (1) A\ast B⊂ A\ast(X-B) and A\ast B≠ A\ast(X-B) (2) X-(A\ast B)=A\ast(X-B) (3) A\ast(X-B)⊂ A\ast B and A\ast(X-B)≠ A\ast B (4) A\ast B=A\ast(X-B)

Q.

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 having 3 elements. Let each element of S belongs to exactly 10 of $A_i^{\prime }$ s and exactly 9 of $B_i^{\prime }$ s. Then, find the value of n.

Q.

If $A\cap B^{\prime }=\Phi ,$ show that $A=A\cap B$ and hence show that $A\subseteq B.$

Q. If ∪ = {1, 3, 5, 7, 9, 11, 13}, then which of the following are subsets of U. B = {2, 4} A = {0} C = {1, 9, 5, 13} D = {5, 11, 1} E = {13, 7, 9, 11, 5, 3, 1} F = {2, 3, 4, 5

Q. If $A=\{x,x\in Z\text{and}\frac{x^2-4x+3}{x^2-8x+15}\le 0\}$, then $n\left(A\right)=$

Q.

If A and B are any two sets, then A ∪ (A ∩ B) is equal to

1. A

2. B

3. A^C

4. B^C

Q. If $A=\{1,2,\{3,4\left\}\right\}$, then the number of elements in $P\left(P\right(A\left)\right)$ is
(where $P\left(A\right)$ is the power set of $A)$

Q.

If S and T are two sets such that S has 21 elements T has 32 elements and $S\cap T$ has 11 elements, how many elements does $S\cup T$ have?

Q.

If A has 3 elements and P(A) denotes the power set of A, then the number of elements in P(P(A)) is

1. 8

2. 9

3. 81

4. 256

Q. If A = [1, 2, 3], B = [4, 5, 6], which of the following are relations from A to B? Give reasons in support of your answer.

(i) [(1, 6), (3, 4), (5, 2)]
(ii) [(1, 5), (2, 6), (3, 4), (3, 6)]
(iii) [(4, 2), (4, 3), (5, 1)]
(iv) A × B.

Q. Let U be the universal set and $A\cup B\cup C=U$. Then $\left[\right(A-B)\cup (B-C)\cup (C-A){]}^c$ is equal to

1. $A\cup B\cup C$
2. $A\cup (B\cap C)$
3. $A\cap B\cap C$
4. $A\cap (B\cup C)$

Q. If X and Y are two sets such that $X\cup Y$ has $18$ elements, X has $8$ elements and Y has $15$ elements; how many elements does $X\cap Y$ have?

Q. Select the pair of disjoint sets.

1. $\{1,2,3,4\}$
2. $\{2,3,4,5\}$
3. $\{3,4,5,6\}$
4. $\{5,6,7,8\}$
5. $\{6,7,8,1\}$
6. $\{1,2,7,8\}$

Q.

If A has 3 elements and P(A) denotes the power set of A, then the number of elements in P(P(A)) is

1. 8

2. 9

3. 81

4. 256

Q. FIND THE NUMBER OF WORDS WITH O MEANING WHICH CAN BE MADE USING ALL LETTERS OF THE WORD FLOWER.IF THESE WORDS ARE TO BE WRITTEN IN DICTIONARY, WHAT IS THE NUMBER OF THE WORD FLOWER.

Q. If $S$ and $T$ are two sets such that $S$ has $21$ elements, $T$ has $32$ elements, and $S\cap T$ has $11$ elements, how many elements does $S\cup T$ have?

Q. If $A=\left\{\right(a,b):a^2+b^2=25\text{and}a,b\in N\}$ then $n\left(A\right)=$

1. $1$
2. $2$
3. $4$
4. $12$

Q. $A$ and $B$ are two finite sets, such that $A$ has $p$ elements and $B$ has $q$ elements. The number of elements in the power set of $A$ is $48$ more than number of elements in the power set of $B.$ Then select the correct statement about $p\&q.$

1. $p=4$
2. $p=6$
3. $q=6$
4. $q=4$

Q. using the property of sets prove that c-b is a subset of c-a, if a is a subset of b