Set Theory Test 7

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Set Theory Test 7

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Weekly Quiz Competition

Question 1
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 2
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 3
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 4
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 5
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

$$P(A)=0.5$$
$$P(B)=0.2$$
$$P(AB)=0.1$$
$$(a)$$ $$P(A\cup B)=P(A)+P(B)-P(A)P(B)$$
$$=[0.5+0.2-(0.5)(0.2)]$$
$$\Rightarrow 0.7-0.1$$
$$\Rightarrow 0.6$$
$$(b)$$ $$P(\bar A \cup B)=(P(B)-P(A\cap B))$$
$$P(A\cup B)=P(A)+P(B)-P((A\cap B))$$
$$\Rightarrow 0.3=0.5+0.2-P(A\cap B)$$
$$P(A\cap B)=0.1$$
$$\because P(\bar A \cap B)=0.2-0.1$$
$$=0.1$$
Question 6
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 7
1 / -0

If A and B are any two sets, then $$ A \cup B$$ is equal to:

A
$$(A-B) \cup (B-A) \cup (A \cap B)$$

B
$$ (A^0 \cap B^0)^0$$

C
$$(A-B) \cup (B-A)$$

D
$$A \cup (B-A)$$

Solution

A & B are two sets
AᴜB = (A-B) + (A∩B) + (B-A)
Question 8
1 / -0

The set $$\left( {A \cap B'} \right)' \cup \left( {B \cap C} \right)$$ is equal to :

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

B
$$A' \cup B$$

C
$$A' \cup C'$$

D
$$A' \cap B$$

Solution
Question 9
1 / -0

If $$n(A \cup B)=8, n(A)=6, n(B)=4$$, then $$n(A\cap B)$$=

A
$$2$$

B
$$4$$

C
$$6$$

D
$$8$$

Solution

$$n(A\cup B)=8\\ n(A)=6\quad ,\quad n(B)=4$$

we know that
$$n(A\cup B)=n(A)+n(B)-n(A\cap B)\\$$
$$ 8=6+4-n(A\cap B)\\$$
$$ n(A\cap B)=10-8=2$$
Question 10
1 / -0

Let $$A, B$$ are two sets such that $$n(A)=6, n(B)=8$$ then the maximum number of elements in $$n(A\cup B)$$ is _________

A
$$7$$

B
$$9$$

C
$$14$$

D
$$None$$

Solution

$$n(A)=6,n(B)=8$$
$$n(A\cup B)$$ will be maximum when $$A$$ & $$B$$ are disjoint sets.
$$A\cap B=\phi \\ n(A\cap B)=0\\ n(A\cup B)=n(A)+n(B)-n(A\cap B)\\ =6+8-0\\ =14\\ n(A\cup B)=14$$

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Re-Attempt Weekly Quiz Competition

Set Theory Test 7

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Set Theory Test 7

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SHARING IS CARING

Question 1

$$30$$

$$32$$

$$20$$

none of these

Solution

Question 2

$$A'$$

$$B$$

$$\phi$$

none of these

Solution

Question 3

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

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

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

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

Solution

Question 4

$$8$$

$$9$$

$$10$$

$$11$$

Solution

Question 5

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

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

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

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

Solution

Question 6

$${2}^{80}$$

$${80}^{2}$$

$$81$$

$$79$$

Solution

Question 7

$$(A-B) \cup (B-A) \cup (A \cap B)$$

$$ (A^0 \cap B^0)^0$$

$$(A-B) \cup (B-A)$$

$$A \cup (B-A)$$

Solution

Question 8

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

$$A' \cup B$$

$$A' \cup C'$$

$$A' \cap B$$

Solution

Question 9

$$2$$

$$4$$

$$6$$

$$8$$

Solution

Question 10

$$7$$

$$9$$

$$14$$

$$None$$

Solution

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