Set Theory Test

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Set Theory Test - 1

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Weekly Quiz Competition

Question 1
3 / -1

Let R be a non-empty relation on a collection of sets defined by ARB if and only if A ∩ B = Ø
Then (pick the TRUE statement)

A
R is relexive and transitive

B
R is symmetric and not transitive

C
R is an equivalence relation

D
R is not relexive and not symmetric

Solution

The correct answer is B asLet A={1,2,3
B={4,5}
C={1,6,7}
now
A∩B=∅
B∩C=∅
B∩C=∅ but
A∩C≠∅
R is not transitive.
A∩A=A
R is not reflexive.
A∩B=B∩A
R is symmetric
So,
A is false as
R is not reflexive or transitive
B is true.
C is false because
R is not transitive or reflexive
D is false because
R is symmetric
Question 2
3 / -1

The binary relation S = Φ (empty set) on set A = {1, 2,3} is

A
neither reflexive nor symmetric

B
symmetric and relexive

C
transitive and relexive

D
transitive and symmetric
Solution
Explanation:
Reflexive : A relation is reflexive if every element of set is paired with itself. Here none of the element of A is paired with themselves, so S is not reflexive.
Symmetric : This property says that if there is a pair (a, b) in S, then there must be a pair (b, a) in S. Since there is no pair here in S, this is trivially true, so S is symmetric.
Transitive : This says that if there are pairs (a, b) and (b, c) in S, then there must be pair (a,c) in S. Again, this condition is trivially true, so S is transitive.
Thus, option (D) is correct.
Question 3
3 / -1

Which of the following sets are null sets ?

A
{0}

B
ø

C
{ }

D
Both (b) & (c)

Solution

There are some sets that do not contain any element at all. For example, the set of months with 32 days. We call a set with no elements the null or empty set. It is represented by the symbol { } or Ø.
Question 4
3 / -1

Number of subsets of a set of order three is

A
3

B
6

C
8

D
9

Solution

Number of subset = 2ⁿ
order 3 = 2³
⇒ 8
Question 5
3 / -1

"n/m" means that n is a factor of m, then the relation T is

A
relexive and symmetric

B
transitive and symmetric

C
relexive, transitive and symmetric

D
relexive, transitive and not symmetric

Solution

′/′ is reflexive since every natural number is a factor of itself that in n/n for n∈N.
′/′ is transitive if n is a factor of m and m is a factor of P, then n is surely a factor of P.
However, ′/′ is not symmetric.
example, 2 is a factor of 4 but 4 is not a factor of 2.
Question 6
3 / -1

The number of elements in the Power set P(S) of the set S = [ [ Φ] , 1, [ 2, 3 ]] is

A
2

B
4

C
8

D
none of these
Question 7
3 / -1

If A and B are sets and A∪ B= A ∩ B, then

A
A = Φ

B
B = Φ

C
A = B

D
None of these
Question 8
3 / -1

Let S be an infinite set and S₁, S₂, S₃, ..., S_n be sets such that S₁ ∪S₂ ∪S₃∪ .......S_n = S then

A
atleast one of the sets S_i is a finite set

B
not more than one of the set S_i can be inite

C
atleast one of the sets S_i is an ininite set

D
none of these

Solution

Let S = S1 ∪ S2 ∪ S3 ∪ .... Sn .
For S to be infinite set, atleast one of sets S_i must be infinite,
if all S_i were finite, then S will also be finite.
Question 9
3 / -1

If X and Y are two sets, then X ∩ (Y ∪ X) C equals

A
X

B
Y

C
Ø

D
None of these

Solution
Question 10
3 / -1

If f : X -> Y and a, b ⊆ X, then f (a ∩ b) is equal to

A
f(a) - f(b)

B
f(a) ∩ f(b)

C
a proper subset of f(a) ∩ f(b)

D
f(b) - f(a)

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Set Theory Test - 1

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Set Theory Test - 1

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Question 1

R is relexive and transitive

R is symmetric and not transitive

R is an equivalence relation

R is not relexive and not symmetric

Solution

Question 2

neither reflexive nor symmetric

symmetric and relexive

transitive and relexive

transitive and symmetric

Solution

Question 3

{0}

ø

{ }

Both (b) & (c)

Solution

Question 4

3

6

8

9

Solution

Question 5

relexive and symmetric

transitive and symmetric

relexive, transitive and symmetric

relexive, transitive and not symmetric

Solution

Question 6

2

4

8

none of these

Question 7

A = Φ

B = Φ

A = B

None of these

Question 8

atleast one of the sets Si is a finite set

not more than one of the set Si can be inite

atleast one of the sets Si is an ininite set

none of these

Solution

Question 9

X

Y

Ø

None of these

Solution

Question 10

f(a) - f(b)

f(a) ∩ f(b)

a proper subset of f(a) ∩ f(b)

f(b) - f(a)

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atleast one of the sets S_i is a finite set

not more than one of the set S_i can be inite

atleast one of the sets S_i is an ininite set