Test: Set Theory- 1 - CAT MCQ

The correct answer is B 

Let, A={1,2,3}
B={4,5} 
C={1,6,7}

now, A∩B=∅ 
B∩C=∅ but  A∩C≠∅
Relation 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

Option D is correct.

• 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.

Set A is Irreflexive, Symmetric, Anti Symmetric, Asymmetric, Transitive.
But it is not Reflexive.

Thus, option (D) is correct.

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.

A set with 'n' elements in it can have '2n' subsets.

eg: Let us consider a set A = {1,2,3}

The possible subsets are:

{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3} and {}

Where, {} is the empty set.

Number of subset = 2ⁿ
order 3 = 2³ ⇒ 8

Therefore, the number of subsets is 8.

′/′ 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. 

The power set of any set S is the set of all subsets of S, including the empty set and S itself, various denoted as p(S).

Let A = {{1,2}, {2,3}, {2,4}}

Power set of A =  {Φ, {{1, 2}}, {{2,3}}, {{2,4}}, {{1,2}, {2,3}}, {{1,2}, {2,4}}, {{2,3}, {2,4}}, {{1,2}, {2,3}, {2,4}}}

Let power set of S = x

Number of elements of power set of x

P(x) = 2ⁿ = 2³ = 8.

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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.

We have X ∩ (Y∪X)^C = X ∩ (Y'∩X') = X ∩ X' ∩ Y' =  Ø ∩ Y = Ø 

Hence Option C.

The only requirement to answer the above question is to know the definition of function- a relation becomes a function if every element in domain is mapped to some element in co-domain and no element is mapped to more than one element.

Now, we have a,b⊆ X. Their intersection can be even empty set. So, lets try out options:

Options a and d don't even need a check.

Lets take a case where a∩b = φ. Now, f(a∩b)=φ, but f(a) ∩ f(b) can be non empty. So, option B can be false.

Option C is always true provided "proper subset" is replaced by "subset". This is because no element in domain of a function can be mapped to more than one element. And the subset needn't be "proper" as for a one-one mapping, we get

f(a∩b) = f(a) ∩ f(b) ,Hence option C