Set Theory Test

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Set Theory Test - 2

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Weekly Quiz Competition

Question 1
3 / -1

The number of elements in the power set of the set {{a, b}, c} is

A
8

B
4

C
3

D
7
Question 2
3 / -1

If R = ((1, 1), (3, 1), (2, 3), (4, 2)), then which of the following represents R2, where R2 is R composite R?

A
((1, 1), (3, 1), (2, 3), (4, 2))

B
f(1, 1), (9, 1), (4, 9), (16, 4))

C
1(1, 3), (3, 3), (3, 4), (3, 2))

D
((1, 1), (2, 1), (4, 3), (3, 1))

Solution

The correct answer is D as
R = ((1, 1), (3, 1), (2, 3), (4, 2))
RoR=R²=((1, 1), (3, 1), (2, 3), (4, 2))((1, 1), (3, 1), (2, 3), (4, 2))
=((1, 1), (3, 1), (2, 1), (4, 3))
take the first set (1,1) then take the second element of this subset check in the other set R is there any starting with 1 if yes then take its second element and make a subset in R² similarly check for all.
like (4,2) (2,3)=(4,3)in R²
Question 3
3 / -1

In a room containing 28 people, there are 18 people who speak English, 15 people who speak Hindi and 22 people who speak Kannada, 9 persons speak both English and Hindi, 11 persons speak both Hindi and Kannada where as 13 persosn speak both Kannada and English. How many people speak all the three languages ?

A
6

B
7

C
8

D
9
Question 4
3 / -1

Order of the power set of a set of order n is

A
n

B
2n

C
n²

D
2ⁿ
Question 5
3 / -1

If f : R ---->R defined by f(x) = x2 + 1, then values of f -1 (17) and f -1(-3) are respectively

A
{Ø}, (4, - 4)

B
{3,-3},{Ø}

C
{Ø},{3,-3}

D
{4,-4},Ø
Question 6
3 / -1

In a beauty contest, half the number of experts voted for Mr. A and two thirds voted for Mr. B. 10 voted for both and 6 did not vote for either. How many experts were there in all ?

A
18

B
36

C
24

D
None of these

Solution

The correct answer is C as
Let,the number of voters (experts) be denoted as x
A/Q
X/2+2x/3-10+6=x
7x/6-4=x
7x-24=6x
x=24
Question 7
3 / -1

Let n(A) denotes the number of elements in set A. If n(A) =p and n(B) = q, then how many ordered pairs (a, b) are there with a ∈ A and b ∈ B ?

A
p²

B
p x q

C
p + q

D
2 pq
Question 8
3 / -1

The set of all Equivalence classes of a set A of cardinality C

A
has the same cardinality as A

B
forms a partition of A

C
is of cardinality 2C

D
is of cardinality C²
Question 9
3 / -1

Let Z denote the set of all integers.
Define f : Z —> Z by
f(x) = {x / 2 (x is even)
0 (x is odd)
then f is

A
onto but not one-one

B
one-one but not onto

C
one-one and onto

D
neither one-one nor-onto
Question 10
3 / -1

Let R be a relation "(x -y) is divisible by m", where x, y, m are integers and m > 1, then R is

A
symmetric but not transitive

B
partial order

C
equivalence relation

D
anti symmetric and not transitive

Solution

a) Since x - x = 0, m
=> x - x is divisible by m
(x,x) ∈ R
=> R is reflexive
b) Let (x,y) ∈ R
=> x - y = mq for some q ∈ I
=> y - x = m(-q)
y - x is divisible by m
(y,x) ∈ R
=> R is symmetric.
c) Let (x,y) and (y,z) ∈ R
=> x - y is divisible by m and y - z is divisible by m
=> x - y = mq and y - z = mq' for some q, q' ∈ I
=>(x-y)+(y-z) = m(q+q')
=> x - z = m(q + q'), q + q' ∈ I
(x,z) ∈ R
=> R is transitive.
Hence the relation is equivalence relation.

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Re-Attempt Weekly Quiz Competition

Set Theory Test - 2

Result

Set Theory Test - 2

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Question 1

8

4

3

7

Question 2

((1, 1), (3, 1), (2, 3), (4, 2))

f(1, 1), (9, 1), (4, 9), (16, 4))

1(1, 3), (3, 3), (3, 4), (3, 2))

((1, 1), (2, 1), (4, 3), (3, 1))

Solution

Question 3

6

7

8

9

Question 4

n

2n

n2

2n

Question 5

{Ø}, (4, - 4)

{3,-3},{Ø}

{Ø},{3,-3}

{4,-4},Ø

Question 6

18

36

24

None of these

Solution

Question 7

p2

p x q

p + q

2 pq

Question 8

has the same cardinality as A

forms a partition of A

is of cardinality 2C

is of cardinality C2

Question 9

onto but not one-one

one-one but not onto

one-one and onto

neither one-one nor-onto

Question 10

symmetric but not transitive

partial order

equivalence relation

anti symmetric and not transitive

Solution

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n²

2ⁿ

p²

is of cardinality C²