Set Theory Test 13

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Set Theory Test 13

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Weekly Quiz Competition

Question 1
1 / -0

The solution set of $$3 x - 4 < 8$$ over the set of non-negative square numbers is

A
$$\displaystyle \left \{ 1,2,3 \right \}$$

B
$$\displaystyle \left \{ 1,4 \right \}$$

C
$$\displaystyle \left \{ 1 \right \}$$

D
$$\displaystyle \left \{ 16 \right \}$$

Solution

$$3x-4<8$$
$$3x<12$$
$$x<4$$
Hence set of non-negative square numbers belonging to the above set is
$$\{1\}$$.
Question 2
1 / -0

Let $$P$$ and $$Q$$ be two sets then what is $$\displaystyle (P\cap Q')\cup (P\cup Q)'$$ equal to ?

A
$$\displaystyle (P\cap Q')\cup (P\cup Q)'=\xi \cap Q'=\xi \cap Q'=\xi $$

B
$$\displaystyle (P\cup Q')\cup (P\cup Q)'=\xi \cap Q'=\xi \cap Q'=\xi $$

C
$$\displaystyle (P\cap Q')\cup (P\cap Q)'=\xi \cap Q'=\xi \cap Q'=\xi $$

D
none of the above

Solution

$$\displaystyle (P\cap Q')\cup (P\cup Q)'=(P\cap Q')\cup (P'\cap Q') $$
$$=$$$$\displaystyle (P\cup P')\cap (P\cup Q')\cap (Q'\cup P')\cap (Q'\cup Q')$$
$$=$$$$\displaystyle \xi \cap \left \{ Q'\cup (P\cap P') \right \}\cap Q'$$
$$=\displaystyle \xi \cap \{Q'\cup \xi )\cap Q'$$
$$=$$$$\displaystyle \xi \cap Q'\cap Q'=\xi \cap Q'=\xi$$
Question 3
1 / -0

If $$A$$ and $$B$$ are finite sets which of the following is the correct statement?

A
$$n(A - B) = n(A) - n(B)$$

B
$$n(A - B) = n(B - A)$$

C
$$n(A - B) = n(A) -$$ $$\displaystyle n\left ( A\cap B \right )$$

D
$$n(A - B) = n(B) - $$$$\displaystyle n\left ( A\cap B \right )$$

Solution
Question 4
1 / -0

U is a universal set and n(U) = 160. A, B and C are subset of U. If n(A) = 50, n(B) = 70, $$\displaystyle n\left ( B\cup C \right )=\Phi $$, $$\displaystyle n\left ( B\cap C \right )=15 $$ and $$\displaystyle A\cup B\cup C=U $$. then n(C) equals

A
40

B
50

C
55

D
60
Question 5
1 / -0

If $$n (A) = 115, n(B) = 326, n(A - B) = 47$$, then $$\displaystyle n(A+B)$$ is equal to

A
373

B
165

C
370

D
None

Solution

$$n(A-B)=47\Rightarrow n(A)-n(A\cap B) = 47\Rightarrow n(A\cap B)=115-47=68$$

Hence $$n(A+B)=115+326-68=373$$
Question 6
1 / -0

If A and B are two disjoint sets and N is the universal set then $$\displaystyle A^{c}\cup \left [ \left ( A\cup B \right )\cap B^{c} \right ]$$ is

A
$$\displaystyle \phi $$

B
A

C
B

D
N

Solution

Since $$A$$ and $$B$$ are disjoint sets $$B^c\cap(A\cup B)$$ will be only $$A$$ as there is no intersection between $$A$$ and $$B$$.

Of course $$A^c\cup A=N$$
Question 7
1 / -0

Suppose $$\displaystyle A_{1},A_{2},....,A_{30}$$ are thirty sets each having 5 elements and $$\displaystyle B_{1},B_{2},....,B_{n}$$ are n sets each with 3 elements. Let $$\displaystyle \bigcup_{i=1}^{30}A_{i} = \bigcup_{j=1}^{n}B_{j}=S $$ and each elements of S belongs to exactly 10 of the $$\displaystyle A_{i}$$ and exactly 9 of the $$\displaystyle B_{j}$$. Then n is equal to-

A
35

B
45

C
55

D
65

Solution

$$A_0,A_1,.............,A_{30}\implies$$ each of 5 elements
$$B_1,B_2,B_3...........n\implies$$ each of 3 elements
The number of elements in the union of the A sets is $$5(30)-r$$ where 'r' is the number repeats likewise the number of elements in the B sets $$3n-rB$$
Each element in the union (in5) is repeated 10 times in A which means if x was the real number of elements in A (not counting repeats) then q out of those 10 should be thrown away or 9x .likewise on the B side 8x of those elements should be thrown away So, $$\implies 150-9x=3n-8x$$
$$n=50-3x$$
Now in figure out what x is we need to use the fact that the union of a group of sets contains every member of each sets . If every element in 'S' is repeated 10 times that means every element in the union of the n's is repeated 10 times .
This means that $$10/10\implies 15$$ is the number of in the A's without repeats counted (same for the B's aswell ) So now
$$\cfrac{50-15}{3}=n$$
$$n=45$$
Subset:- A proper subset is nothing but it contain atleast one more element of main set .
Ex:$$\{3,4,5\}$$ is a set then the possible subsets are
$$\{3\},\{4\},\{5\},\{1,5\},\{3,4\}$$
Question 8
1 / -0

Suppose $$\displaystyle A_{1},A_{2}.....A_{30}$$ are thirty sets having 5 elements and $$\displaystyle B_{1},B_{2}....B_{n}$$ are n sets each with 3 elements. Let $$\displaystyle \bigcup_{i=1}^{30}Ai=\bigcup_{i=1}^{n}Bj=S$$ and each elements of S belongs to exactly 10 of the Ai's and exactly 9 of the Bj's. Then n is equal to

A
35

B
45

C
55

D
65

Solution

Since each element of S belongs to $$10$$ $$ A_{i}s$$, there are only $$15 $$elements in S $$\left[\dfrac{30\times 5}{10}\right]$$

Since each of the $$15$$ elements belongs to 9 $$B_j s$$ there are 135 elements in the second union.

No. of sets $$=\dfrac{135}{3}=45$$
Question 9
1 / -0

S = {1, 2, 3, 5, 8, 13, 21, 34}. Find $$\displaystyle \sum max\left ( A \right )$$ where the sum is taken over all 28 two elements subsets A to S

A
844

B
480

C
484

D
488

Solution
Question 10
1 / -0

Given n(A) = 11, n(B) = 13, n(C) = 16, $$\displaystyle n\left ( A\cap B \right )=3,n\left ( B\cap C \right )=6,n\left ( A\cap C \right )=5\: \: and\: \: n\left ( A\cap B\cap C \right )=2$$ then the value of $$\displaystyle n [ A\cup B \cup C ]=$$

A
$$24$$

B
$$27$$

C
$$25$$

D
$$28$$

Solution

We know,
$$n(A\cup B\cup C)= n(A) +n(B) +n(C)-n(A\cap B)-n(B\cap C) -n(C\cap A) + n(A\cap B \cap C)$$

$$=11+13+16-3-6-5+2=28$$

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Set Theory Test 13

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Set Theory Test 13

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Question 1

$$\displaystyle \left \{ 1,2,3 \right \}$$

$$\displaystyle \left \{ 1,4 \right \}$$

$$\displaystyle \left \{ 1 \right \}$$

$$\displaystyle \left \{ 16 \right \}$$

Solution

Question 2

$$\displaystyle (P\cap Q')\cup (P\cup Q)'=\xi \cap Q'=\xi \cap Q'=\xi $$

$$\displaystyle (P\cup Q')\cup (P\cup Q)'=\xi \cap Q'=\xi \cap Q'=\xi $$

$$\displaystyle (P\cap Q')\cup (P\cap Q)'=\xi \cap Q'=\xi \cap Q'=\xi $$

none of the above

Solution

Question 3

$$n(A - B) = n(A) - n(B)$$

$$n(A - B) = n(B - A)$$

$$n(A - B) = n(A) -$$ $$\displaystyle n\left ( A\cap B \right )$$

$$n(A - B) = n(B) - $$$$\displaystyle n\left ( A\cap B \right )$$

Solution

Question 4

40

50

55

60

Question 5

373

165

370

None

Solution

Question 6

$$\displaystyle \phi $$

A

B

N

Solution

Question 7

35

45

55

65

Solution

Question 8

35

45

55

65

Solution

Question 9

844

480

484

488

Solution

Question 10

$$24$$

$$27$$

$$25$$

$$28$$

Solution

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