Sets Test

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Sets Test - 19

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Weekly Quiz Competition

Question 1
1 / -0

Let $$A$$ and $$B$$ have $$3$$ and $$6$$ elements respectively. What can be the minimum number of elements in $$A\cup B$$?

A
$$3$$

B
$$6$$

C
$$9$$

D
$$18$$

Solution

Ans. $$(b)$$.
$$n\left( A\cup B \right) =n(A)+n(B)-n\left( A\cap B \right)$$
Now $$A$$ has $$3$$ elements and $$B$$ has $$6$$ elements. If they are disjoint, then $$n\left( A\cap B \right) =0$$.
$$\therefore n\left( A\cup B \right) =6+3=9$$
If $$A\subset B$$ then $$A\cup B=B$$
$$\therefore \left( A\cup B \right) =n(B)=6$$
$$B$$ cannot be a subset of $$A$$ and hence the other possibility of $$A\cup B=A$$ is ruled out.
Question 2
1 / -0

$$25$$ people for applied for programme $$A$$, $$50$$ people for programme $$B$$, $$10$$ people for both. So number of employee applied only for $$A$$ is

A
$$15$$

B
$$20$$

C
$$35$$

D
$$40$$

Solution

$$n(A - B) = n(A) - n(A \cap B) = 25 -10 -15$$
Question 3
1 / -0

If $$P(A) = 0.8 , P(B) = 0.5 $$ & $$P(B/A) =0.4 $$ find (i) $$P(A \cap B) $$ (ii) $$P(A/B)$$ (iii) $$P(A\cup B)$$.

A
$$i) 0.32, ii)0.64 , iii)0.98$$

B
$$i) 0.62, ii)0.34 , iii)0.88$$

C
$$i) 0.66, ii)0.32 , iii)0.98$$

D
none

Solution

$$\\\>P(\frac{B}{A})=\>\frac{P(B\>\cap\>A)}{P(A)}\\\>0.4=\frac{P(B\cap\>A)}{0.8}\\$$
$$\therefore\>P(B\cap\>A)=0.32\\$$, $$\implies \>P(A\cap\>B)=0.32\\$$
$$then\>\>P(\frac{A}{B})=\>\frac{P(A\>\cap\>B)}{P(B)}=(\frac{0.32}{0.5})=0.64\>\\$$
$$and\>P(A\cup\>B)=P(A)+P(B)-P(A\cap\>B)\\=0.8+0.5-0.32\\\>=0.98$$
Question 4
1 / -0

Let $$A$$ is a finite set such that $$n(A)=6$$ then $$n[P(A)]$$ is

A
$$32$$

B
$$36$$

C
$$63$$

D
$$64$$

Solution

We have if a set contains $$n$$ element then its power set contains $$2^n$$ elements.
In our problem $$n=6$$.
Then $$n[P(A)]=2^6=64$$.
Question 5
1 / -0

Let $$A$$ and $$B$$ be two sets such that $$A\cap B=\phi$$. Find the value of $$(A\cup B')=$$

A
$$A'$$

B
$$B$$

C
$$\phi$$

D
none of these

Solution

Given, $$A\cap B=\phi$$.
Now,
$$(A\cup B')$$
$$=(A'\cap B)'$$ [ Using De Morgan's law]
$$=B'$$. [ As $$A'\cap B=B-(A\cap B)=B$$ since $$A\cap B=\phi$$]
Question 6
1 / -0

Let $$n(A)=28$$,$$n(A\cap B)=8$$, $$n(A\cup B)=52$$, then $$n(A\cap B')=$$.

A
$$30$$

B
$$32$$

C
$$20$$

D
none of these

Solution

Given $$n(A)=28$$,$$n(A\cap B)=8$$.
We have $$A\cap B'=A-A\cap B$$.
This give $$n(A\cap B')=n(A)-n(A\cap B)$$
or, $$n(A\cap B')=28-8=20$$.
Question 7
1 / -0

If $$n(A)$$ denotes the number of elements in set A and if $$n(A)=4, n(B)=5$$ and $$n(A\cap B)=3$$ then $$n\left[ \left( A\times B \right) \cap \left( B\times A \right) \right] =$$

A
$$8$$

B
$$9$$

C
$$10$$

D
$$11$$

Solution

For $$(A\times B)\cap (B\times A)$$ we have to do the mapping of $$A\times B$$ or $$B\times A$$ between common elements.
no. of ways of mapping will be $$3\times 3=9$$
$$n[(A\times B)\cap(B\times A)]=9$$
Question 8
1 / -0

If two sets $$A$$ and $$B$$ are having $$80$$ elements in common, then the number of element common to each of the sets $$A\times B$$ and $$B\times A$$ are

A
$${2}^{80}$$

B
$${80}^{2}$$

C
$$81$$

D
$$79$$

Solution
Question 9
1 / -0

Find the set of values of x for which it satisfies $$- 2 \le \left[ x \right] \le 4.$$ (where $$\left[ \ \ \right]$$ denotes the greatest integer function )

A
$$\left[ { - 2,\left. 5 \right)} \right.$$

B
$$\left( { - 2,\left. 5 \right]} \right.$$

C
$$\left( { - 2,\left. 5 \right)} \right.$$

D
$$\left[ { - 2,\left. 5 \right]} \right.$$

Solution

Given $$-2\leq[x]\leq 4$$

As we know for$$ \left [ x \right ] \geq n \Rightarrow x\geq n$$ for $$ n\in Z$$
and for $$[x]\leq n \Rightarrow x< n+1$$ for$$ n\in Z$$

$$\Rightarrow [x]\geq -2$$ and $$[x]\leq 4$$
$$\Rightarrow x\geq -2$$ and $$x< (4+1)$$
$$\Rightarrow x\geq -2$$ and $$x< 5$$
$$\Rightarrow -2 \leq x < 5$$

$$\therefore x\in [-2,5)$$
Question 10
1 / -0

Given P(A)=$$0.5,$$P(B)=$$0.2$$ and P(AB)=0.1;find

A
$$P\left( {A \cup B} \right)$$

B
$$P\left( {\overline A \cap B} \right)$$

C
$$P\left( {A \cup \overline B} \right)$$

D
$$P\left( {\overline A \cap \overline B} \right)$$

Solution

$$P(A)=0.5$$
$$P(B)=0.2$$
$$P(AB)=0.1$$
$$(a)$$ $$P(A\cup B)=P(A)+P(B)-P(A)P(B)$$
$$=[0.5+0.2-(0.5)(0.2)]$$
$$\Rightarrow 0.7-0.1$$
$$\Rightarrow 0.6$$
$$(b)$$ $$P(\bar A \cup B)=(P(B)-P(A\cap B))$$
$$P(A\cup B)=P(A)+P(B)-P((A\cap B))$$
$$\Rightarrow 0.3=0.5+0.2-P(A\cap B)$$
$$P(A\cap B)=0.1$$
$$\because P(\bar A \cap B)=0.2-0.1$$
$$=0.1$$

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Sets Test - 19

Result

Sets Test - 19

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SHARING IS CARING

Question 1

$$3$$

$$6$$

$$9$$

$$18$$

Solution

Question 2

$$15$$

$$20$$

$$35$$

$$40$$

Solution

Question 3

$$i) 0.32, ii)0.64 , iii)0.98$$

$$i) 0.62, ii)0.34 , iii)0.88$$

$$i) 0.66, ii)0.32 , iii)0.98$$

none

Solution

Question 4

$$32$$

$$36$$

$$63$$

$$64$$

Solution

Question 5

$$A'$$

$$B$$

$$\phi$$

none of these

Solution

Question 6

$$30$$

$$32$$

$$20$$

none of these

Solution

Question 7

$$8$$

$$9$$

$$10$$

$$11$$

Solution

Question 8

$${2}^{80}$$

$${80}^{2}$$

$$81$$

$$79$$

Solution

Question 9

$$\left[ { - 2,\left. 5 \right)} \right.$$

$$\left( { - 2,\left. 5 \right]} \right.$$

$$\left( { - 2,\left. 5 \right)} \right.$$

$$\left[ { - 2,\left. 5 \right]} \right.$$

Solution

Question 10

$$P\left( {A \cup B} \right)$$

$$P\left( {\overline A \cap B} \right)$$

$$P\left( {A \cup \overline B} \right)$$

$$P\left( {\overline A \cap \overline B} \right)$$

Solution

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