Sets Test

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Sets Test - 30

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Weekly Quiz Competition

Question 1
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 2
1 / -0

If A is a finite set, let $$P(A)$$ denote the set all subsets of A and $$n(A)$$ denote the number of elements in A. If for two finite sets X and Y, $$n[P(X)] = n[P(Y)] + 15$$ then find $$n(X)$$ and $$n(Y)$$

A
$$n(X) = 4; n(Y) = 0$$

B
$$n(X) = 5; n(Y) = 0$$

C
$$n(X) = 6; n(Y) = 0$$

D
$$n(X) = 7; n(Y) = 0$$

Solution

If $$X$$ is a finite set.
Let,
Number of elements in subset $$A$$ be $$n(A)=m$$
Number of elements in subset $$B$$ be $$n(B)=n$$
Then total number of subsets of finitr set containing 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 3
1 / -0

$$A - B =$$ _____

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

B
$$B - A$$

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

D
$$A'\cap B$$

Solution

A-B means everything in A except for anything in $$A \cap B$$
Pick an element x. Let $$ x\in(A-B)$$
$$\therefore x \in A \; but \; x \notin B$$
$$\therefore x\in A , x\in B'$$
$$x\in (A \cap B') = A- (A \cap B)$$
or, $$x\in (A- B)$$
Hence, $$A-B = (A \cap B') = A- (A \cap B)$$
Question 4
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 5
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 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

Let $$S = \left \{(a, b, c)\epsilon N\times N\times N : a + b + c = 21. a \leq b\leq c\right \}$$ and $$T = \left \{a, b, c)\epsilon N\times N\times N : a, b, c,\ are\ in\ A.P.\right \}$$, where $$N$$ is the set of all natural numbers. Then the number of elements in the set $$S\cap T$$ is

A
$$6$$

B
$$7$$

C
$$13$$

D
$$14$$

Solution

$$a + b + c = 21$$ and $$b = \dfrac {a + c}{2}$$
$$\Rightarrow a + c = 14$$ and $$b = 7$$
So, a can take values from $$1$$ to $$6$$, when $$c$$ ranges from $$13$$ to $$8$$, or $$a = b = c = 7$$
So, $$7$$ triplets
Question 10
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

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Sets Test - 30

Result

Sets Test - 30

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SHARING IS CARING

Question 1

None

II only

I and II only

I and III only

I, II, and III

Solution

Question 2

$$n(X) = 4; n(Y) = 0$$

$$n(X) = 5; n(Y) = 0$$

$$n(X) = 6; n(Y) = 0$$

$$n(X) = 7; n(Y) = 0$$

Solution

Question 3

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

$$B - A$$

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

$$A'\cap B$$

Solution

Question 4

$$B$$

$$A-B$$

$$A\cap B$$

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

Solution

Question 5

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

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

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

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

Solution

Question 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

$$6$$

$$7$$

$$13$$

$$14$$

Solution

Question 10

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

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

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

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

Solution

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