Set Theory Test 16

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Set Theory Test 16

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Weekly Quiz Competition

Question 1
1 / -0

Three sets $$A, B, C$$ are such that $$\displaystyle A=B\cap C$$ and $$\displaystyle B=C\cap A$$, then

A
$$\displaystyle A\subset B$$

B
$$\displaystyle A\supset B$$

C
$$\displaystyle A\equiv B$$

D
$$\displaystyle A\subset { B }^{ \prime }$$

Solution

Since, $$\displaystyle A=B\cap C$$ and $$\displaystyle B=C\cap A$$, then $$\displaystyle A\equiv B$$
Question 2
1 / -0

If n is a member of both set A$$=\left\{\displaystyle\frac{4}{7}, 1, \frac{5}{2}, 4, \frac{1}{2}, 7\right\}$$ and set B$$=\left\{\displaystyle\frac{4}{7}, \frac{7}{4}, 4, 7\right\}$$, which of the following must be true?
I. n is an integer.
II. $$4n$$ is an integer.
III. $$n=4$$

A
None

B
II only

C
I and II only

D
I and III only

E
I, II, and III

Solution

None of the criteria satisfy all the elements in the intersection.

$$\dfrac47$$ is not an integer nor is $$\dfrac {4\times 4}{7}$$ an integer.

Of course 4 is not the only element in the intersection. Hence 3 is untrue.
Question 3
1 / -0

If the universal set $$\{x\in W ,3<x≤12\} ,A=\{5,7,9\}$$, then $$A'=$$

A
$$\{3,6,8,10,11,12\}$$

B
$$\{4,6,8,10,11,12\}$$

C
$$\{6,8,10,11,12\}$$

D
None of the above

Solution

Given universal set is $$\{x\in W ,3<x≤12\}$$
It can be written as a set $$\{4,5,6,7,8,9,10,11,12\}$$
$$A$$ is given as $$A=\left\{5,7,9\right\}$$
$$\therefore A'=W-A=$$ All the elements in the universal set but not in set $$A$$
$$ =\{4,6,8,10,11,12\}$$
Hence, option B is correct
Question 4
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
$$\left \{a, c, e, f, g, h, i, j\right \}$$

B
$$\left \{a, c, d, e, f, h, i, j\right \}$$

C
$$\left \{a, c, d, e, f, g, h, j\right \}$$

D
$$\left \{a, c, d, e, f, g, h, i, j\right \}$$

Solution
Question 5
1 / -0

For any two sets $$A$$ and $$B$$, $$A-\left( A-B \right) $$ equals

A
$$B$$

B
$$A-B$$

C
$$A\cap B$$

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

Solution

Now, $$A-\left( A-B \right) =A-\left( A\cap { B }^{ C } \right) $$
$$=A\cap { \left( A\cap { B }^{ C } \right) }^{ C }$$
$$=A\cap \left( { A }^{ C }\cup B \right) $$
$$=\left( A\cap { A }^{ C } \right) \cup \left( A\cap B \right) $$
$$=A\cap B$$
Question 6
1 / -0

In a class, $$20$$ opted for Physics, $$17$$ for Maths, $$5$$ for both and $$10$$ for other subjects. The class contains how many students?

A
$$35$$

B
$$42$$

C
$$52$$

D
$$60$$

Solution

$$n(P) = 20$$

$$n(M) = 17$$

$$n(M \cap P) = 5$$

$$n(other \; subjects)=10$$

$$n(M\cup P) =n(M)+n(P)-n(M\cap P)$$

$$n(M\cup P) =17+20-5 = 32$$

Total students $$= n(P\cup M)+n (other \; subjects )$$

$$32+10=42$$
Question 7
1 / -0

If $$A = \{ a, b, p, d\} B = \{ p, d, e\} C = \{p, e, f, g\}$$ then find

$$A \times (B \cap C ) $$ is equal to

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

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

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

D
none

Solution
Question 8
1 / -0

In a class of 250 students, 175 take mathematics and 142 take science. How many take both mathematics

and science? (All take math and/or science.)

A
67

B
75

C
33

D
184

E
cannot be determined from information given

Solution
Question 9
1 / -0

If X is a finite set. Let $$P(X)$$ denote the set of all subsets of X and let $$n(X)$$ denote the number of elements in X. If for two finite subsets $$A, B, n(P(A)) = n(P(B)) + 15$$ then $$n(B) = $$ ____, $$n(A) =$$ _____

A
$$n(A) = 4, n(B) = 0$$

B
$$n(A) = 5, n(B) = 0$$

C
$$n(A) = 4, n(B) = 2$$

D
$$n(A) = 6, n(B) = 0$$

Solution

If X is a finite set.
Let,
Number of element in a subset $$A$$ be $$n(A)$$ =$$m$$ and
Number of elements in subset $$B$$ be $$n(B)$$=$$n$$
Then total number of subsets of finite set contaning say $$n$$ elements is $$2^n$$.
Therefore,
$$n[P(A)]=2^m$$, $$n[P(B)]=2^n$$
Substituting in equations , we get:-
$$2^m=2^n+15$$
$$2^m-2^n=15$$

$$m=4,n=0$$

Option $$\textbf A$$ is correct
Question 10
1 / -0

P, Q and R are three sets and $$\xi = P\cup Q\cup R$$. Given that $$n(\xi) = 60, n (P\cap Q) = 5, n(Q\cap R) = 10, n(P) = 20$$ and $$n(Q) = 23$$, find $$n(P\cup R)$$

A
$$37$$

B
$$38$$

C
$$45$$

D
$$52$$

Solution

$$n(U) =60 (Universal \; Set) $$

$$n(P \cap Q ) = 5$$

$$n(Q \cap R ) = 10$$

$$n(P ) = 20$$

$$n(Q)=23$$

$$n(P \cup Q ) = n(P) + n(Q)- n(P \cap Q ) = 20+23-5 = 38$$

$$n(R)=n(U)-n(P \cup Q ) +n(Q\cap R)= 60-38+10 = 32$$

$$n(P \cup R ) = n(P) + n(R)- n(P \cap R ) = 20+32-0 =52$$

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Set Theory Test 16

Result

Set Theory Test 16

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SHARING IS CARING

Question 1

$$\displaystyle A\subset B$$

$$\displaystyle A\supset B$$

$$\displaystyle A\equiv B$$

$$\displaystyle A\subset { B }^{ \prime }$$

Solution

Question 2

None

II only

I and II only

I and III only

I, II, and III

Solution

Question 3

$$\{3,6,8,10,11,12\}$$

$$\{4,6,8,10,11,12\}$$

$$\{6,8,10,11,12\}$$

None of the above

Solution

Question 4

$$\left \{a, c, e, f, g, h, i, j\right \}$$

$$\left \{a, c, d, e, f, h, i, j\right \}$$

$$\left \{a, c, d, e, f, g, h, j\right \}$$

$$\left \{a, c, d, e, f, g, h, i, j\right \}$$

Solution

Question 5

$$B$$

$$A-B$$

$$A\cap B$$

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

Solution

Question 6

$$35$$

$$42$$

$$52$$

$$60$$

Solution

Question 7

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

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

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

none

Solution

Question 8

67

75

33

184

cannot be determined from information given

Solution

Question 9

$$n(A) = 4, n(B) = 0$$

$$n(A) = 5, n(B) = 0$$

$$n(A) = 4, n(B) = 2$$

$$n(A) = 6, n(B) = 0$$

Solution

Question 10

$$37$$

$$38$$

$$45$$

$$52$$

Solution

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