Set Theory Test 4

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Set Theory Test 4

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Question 1
1 / -0

Let $$U=\{x: \in, W: 3<x< 12\} $$, $$B=\{4,6,8,10\}$$ . $$B'$$

A
$$\{6,7,9,11,12\}$$

B
$$\{5,7,9,11\}$$

C
$$\{5,7,9,10,11,12 \}$$

D
None of the above

Solution

Given universal set is $$\{x\in W ,3<x<12\}$$
It can be written as a set $$\{4,5,6,7,8,9,10,11\}$$
$$B$$ is given as $$B=\left\{4,6,8,10\right\}$$
$$\therefore B'=W-B=$$ All the elements in the universal set but not in set $$B$$
$$ =\{5,7,9,11\}$$
Hence, option B is correct
Question 2
1 / -0

Directions For Questions

$$\mu = \left \{a, b, c, d, e, f, g, h, i, j\right \}$$
$$P = \left \{a, b, c, e\right \}$$
$$Q = \left \{b, c, d, f\right \}$$ and
$$R = \left \{c, f, h, i, j\right \}$$
Find the number of elements of the set

...view full instructions

$$P\cap Q\cup R$$

A
$$\left \{b, c, d, h, i, j\right \}$$

B
$$\left \{b, c, d, g, i, j\right \}$$

C
$$\left \{b, c, f, h, i, j\right \}$$

D
$$\left \{b, c, f, g, i, j\right \}$$

Solution

$$ P = \left \{ a,b,c,e \right \} $$
$$ Q = \left \{ b,c,d,f \right \} $$
$$ \Rightarrow (P\cap Q) = \left \{ b,c \right \} $$
As $$ R = \left \{ c,f,h,i,j\right \} $$
$$ \Rightarrow \boxed {(P\cap Q)\cup R = \left \{ b,f,h,i,j \right \}} $$
Question 3
1 / -0

Given $$P(A \cup B)=0.6, P(A\cap B)=0.2$$, the probability of exactly one of the event occurs is

A
$$0.4$$

B
$$0.2$$

C
$$0.6$$

D
$$0.8$$

Solution

Given, $$P(A\cup B)=0.6, P(A\cap B)=0.2$$
Probability of exactly one of the event occurs is $$P(\bar{A}\cap B)+P(A\cap \bar{B})$$
$$=P(B)-P(A\cap B)+P(A)-P(A\cap B)$$
$$=P(A\cup B)+P(A\cap B)-2P(A\cap B)$$
$$[\because P(A\cup B)=P(A)+P(B)-P(A\cap B)]$$
$$=P(A\cup B)-P(A\cap B)$$
$$=0.6-0.2$$
$$=0.4$$
Question 4
1 / -0

If $$A, B$$ and $$C$$ are any three set, then $$A \cup (B\cap C) =$$

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

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

C
$$ (A \cap B) \cap (A\cap C)$$

D
none

Solution

Using distributive law of sets option B is correct
Or it is the distributive law itself
Question 5
1 / -0

If $$A = \left \{1, 2, 3, 4\right \}$$, what is the number of subsets of A with at least three elements?

A
$$3$$

B
$$4$$

C
$$5$$

D
$$10$$

Solution

A subset containing $$3$$ elements $$= \left \{1, 2, 3\right \}; \left \{1, 3, 4\right \}; \left \{1, 2, 4\right \}$$ and $$\left \{2, 3, 4\right \}$$
A subset containing $$4$$ elements $$= \left \{1, 2, 3, 4\right \}$$
$$\therefore$$ there are five subsets containing at least $$3$$ elements.
Question 6
1 / -0

The set of integers is closed with respect to which one of the following?

A
Addition only

B
Multiplication only

C
By addition and multiplication

D
Division

Solution

From group theory, integers are closed w.r.t. both addition & multiplication
Question 7
1 / -0

In the Venn diagram, the numbers represent the number of elements in the subsets. Given that $$\xi = F\cup G\cup H$$ and $$n(\xi) = 42$$, find $$n(G'\cup H)$$

A
$$18$$

B
$$28$$

C
$$30$$

D
$$38$$

Solution

$$n(\xi) = 42$$

$$n(G) = 12, n(F) = 17, n(H) = 8$$

$$n(F \cap G) = 5, n(G \cap H) = 0, n(F \cap H) = 0$$

$$n(G’)= 42-7-5 =30$$

$$n(H)= 8, n(G’ \cup H) = n(G')+n(H)-n(G'\cap H)=30+8-8 = 30$$
Question 8
1 / -0

Directions For Questions

$$\mu = \left \{a, b, c, d, e, f, g, h, i, j\right \}$$
$$P = \left \{a, b, c, e\right \}$$
$$Q = \left \{b, c, d, f\right \}$$ and
$$R = \left \{c, f, h, i, j\right \}$$
Find the number of elements of the set

...view full instructions

$$(P\cap Q)\cup (Q\cap R)$$

A
$$\left \{b, c, f\right \}$$

B
$$\left \{b, c, d\right \}$$

C
$$\left \{b, c, d, f\right \}$$

D
$$\left \{b, c, f, g\right \}$$

Solution

$$ P = \left \{ a,b,c,e \right \};Q = \left \{ b_{n}c_{n}d_{n}f \right \} $$
$$ \Rightarrow (P\cap Q) = \left \{ b,c \right \} $$
As $$ R = \left \{ c,f,n,i,j \right \} \Rightarrow (Q\cap R) = \left \{ c,f \right \} $$
$$ \boxed { (P\cap Q)\cup (Q\cap R) = \left \{ b,c,f \right \}} $$
Question 9
1 / -0

In a school with an envolment of $$950$$ students, each student must join either the lions club or the country club or both. Given that $$646$$ students are members of the lions club and $$532$$ are members of the country club, calculate the number of students who are members of both clubs

A
$$228$$

B
$$230$$

C
$$232$$

D
$$234$$

Solution

Total number of students $$=950$$
Number of students who are members of lion club $$=646$$
Number of students who are members of country club $$=532$$
Number of students who are members of both club $$=(646+532)-950$$
$$=1178-950$$
$$=228$$
$$\therefore$$ There are $$228$$ students who joined both clubs.
Question 10
1 / -0

If A and B be two sets containing $$4$$ and $$8$$ elements respectively, what can be the maximum number of elements in $$A\cup B$$? Find also, the minimum number of elements in $$(A\cup B)$$?

A
Maximum number of elements $$= 12$$
Minimum number of elements $$= 8$$

B
Maximum number of elements $$= 14$$
Minimum number of elements $$= 8$$

C
Maximum number of elements $$= 12$$
Minimum number of elements $$= 9$$

D
Maximum number of elements $$= 14$$
Minimum number of elements $$= 7$$

Solution

$${\textbf{Step -1: Find maximum number of elements.}}$$
$${\text{Given,}}$$
$$n(A) =4$$
$$n(B) =8$$
$${\text{Minimum number of elements in}}$$ $$A\cup B=8.$$
$${\text{A is subset of B.}}$$
$${\text{Maximum number of elements in}}$$ $$A\cup B=4+8=12.$$
$${\textbf{[All element taken in both set A and B and elements in both sets are different.]}}$$
$${\textbf{Hence , maximum number of elements in}}$$ $$\mathbf{(A \cup B) =12.}$$

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Set Theory Test 4

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Question 1

$$\{6,7,9,11,12\}$$

$$\{5,7,9,11\}$$

$$\{5,7,9,10,11,12 \}$$

None of the above

Solution

Question 2

$$\left \{b, c, d, h, i, j\right \}$$

$$\left \{b, c, d, g, i, j\right \}$$

$$\left \{b, c, f, h, i, j\right \}$$

$$\left \{b, c, f, g, i, j\right \}$$

Solution

Question 3

$$0.4$$

$$0.2$$

$$0.6$$

$$0.8$$

Solution

Question 4

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

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

$$ (A \cap B) \cap (A\cap C)$$

none

Solution

Question 5

$$3$$

$$4$$

$$5$$

$$10$$

Solution

Question 6

Addition only

Multiplication only

By addition and multiplication

Division

Solution

Question 7

$$18$$

$$28$$

$$30$$

$$38$$

Solution

Question 8

$$\left \{b, c, f\right \}$$

$$\left \{b, c, d\right \}$$

$$\left \{b, c, d, f\right \}$$

$$\left \{b, c, f, g\right \}$$

Solution

Question 9

$$228$$

$$230$$

$$232$$

$$234$$

Solution

Question 10

Maximum number of elements $$= 12$$Minimum number of elements $$= 8$$

Maximum number of elements $$= 14$$Minimum number of elements $$= 8$$

Maximum number of elements $$= 12$$Minimum number of elements $$= 9$$

Maximum number of elements $$= 14$$Minimum number of elements $$= 7$$

Solution

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Maximum number of elements $$= 12$$
Minimum number of elements $$= 8$$

Maximum number of elements $$= 14$$
Minimum number of elements $$= 8$$

Maximum number of elements $$= 12$$
Minimum number of elements $$= 9$$

Maximum number of elements $$= 14$$
Minimum number of elements $$= 7$$