Sets Test

Result

Sets Test - 45

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Weekly Quiz Competition

Question 1
1 / -0

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

A
15

B
3

C
45

D
35

Solution

If elements are not repeated then the number of elements in $$A_1\cup A_2\cup A_3\cup A_{30}$$ is $$30\times 5$$
But each elements is used 10 times
so, $$S=\cfrac{30\times 5}{10}=15$$
If the elements in $$B_1,B_2,.....,B_n$$ are not repeated.
then total number of element in
$$B_1\cup B_2\cup B_3.....\cup B_n$$ is 3n
but each elements is repeated 9 times
so,
$$S=\cfrac{3n}{9}=15=\cfrac{3n}{9}$$
$$n=45$$
Question 2
1 / -0

The set $$(A\cap B')'\cup (B\cap C)$$ is equal to?

A
$$A'\cup B\cup C$$

B
$$A'\cup B$$

C
$$A'\cup C'$$

D
$$A'\cap B$$

Solution

$$\left( A\cap B' \right) '\cup \left( B\cap C \right) $$
$$=\left( A\cap \bar B \right)\cup \left( B\cap C \right) $$
$$=\left( \bar A\cap B\right)\cup \left( B\cap C \right) $$
$$=\bar A\cup B\cup C$$
Hence, the answer is $$ A'\cup B\cup C.$$
Question 3
1 / -0

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

A
$$B \cap {C^c}$$

B
$${B^c} \cap {C^c}$$

C
$$B \cap {C}$$

D
$$A \cap {B^c} \cap {C^c}$$

Solution
Question 4
1 / -0

If $$A = \{ 4,5,8,12 \} , B = \{ 1,4,6,9 \} \text { and } C = \{ 1,2,3,4 \}$$ then $$A - ( C - B ) =$$

A
$$A$$

B
$$B$$

C
$$C$$

D
$$\phi$$
Question 5
1 / -0

The number of subsets $$R\,of\,P\, = \left\{ {1,2,3,....8} \right\}$$ which satisfies the property 'There exist integers $$a < \,b < \,c$$ with $$a\, \in \,R,b\, \notin \,R,c\, \in \,R''\,$$ is

A
215

B
219

C
222

D
223

Solution
Question 6
1 / -0

Given $$n(U) =20, n(A) =12, n(B) =9, n(A \cap B) =4$$, where U is the universal set, A and B are subset of U, then $$n((A \cup B)^C)=$$

A
$$17$$

B
$$9$$

C
$$11$$

D
$$3$$

Solution
Question 7
1 / -0

if S is a set of p(x) is polynomial of degree <2 such that p(0)=, P(1)=1, p(x)>0 $$\forall \quad x\quad \varepsilon $$ (0, 1) then

A
S=0

B
$$S=ax+91-1){ x }^{ 2 }\forall a\varepsilon (0,\infty )$$

C
$$S=ax+(1-a){ x }^{ 2 }\forall a\varepsilon R$$

D
$$S=ax+(1-a){ x }^{ 2 }\forall a\varepsilon R(0,\quad 2)$$

Solution
Question 8
1 / -0

X and Y are two sets and $$f:X\rightarrow Y$$. If $$f(c)=\left\{ y;c\subset X,y\subset Y \right\} $$ and $${ f }^{ 1 }(d)=\left\{ x;d\subset Y,x\subset X \right\} $$, then the true statement is

A
$$f({ f }^{ 1 }(b))=b$$

B
$$f({ f }^{ 1 }(a))=a$$

C
$$f({ f }^{ 1 }(b))=b,b\subset y$$

D
$$f({ f }^{ 1 }(a))=a,a\subset x$$
Question 9
1 / -0

A = {n/n is a digits in the number 33591} and $$B=\left\{ n/n\in N,n<10 \right\} ,$$ then $$B-A = $$

A
$$\left\{ 2,4,6,8 \right\} $$

B
$$\left\{ 7,2,4,8,6 \right\} $$

C
$$\left\{ 1,3,5,7 \right\} $$

D
$$\left\{ \left( 1,2 \right) ,\left( 1,3 \right) ,\left( 2,3 \right) \right\} $$

Solution
Question 10
1 / -0

Let $$A,B,C$$ finite sets. Suppose then $$n(A)=10, n(B)=15, n(C)=20, n(A\cap B)=8$$ and $$n(B\cap C)=9$$. Then the possible value of $$n(A\cup B\cup C)$$ is

A
$$26$$

B
$$27$$

C
$$28$$

D
Any of the three values $$26, 27, 28$$ is possible

Solution