SET Practice DS SET – 1
DS: Set Theory – Level 1
Directions: Choose the correct option (A-E) based on sufficiency.
1.
What is the number of elements in \( A \cup B \)?
(1) Set A has 10 elements.
(2) Set B has 15 elements, and 5 elements are in both A and B.
(1) Set A has 10 elements.
(2) Set B has 15 elements, and 5 elements are in both A and B.
Solution (C):
Formula: \( n(A \cup B) = n(A) + n(B) – n(A \cap B) \).
(1) Gives \( n(A) \). Missing \( n(B) \) and intersection. Not sufficient.
(2) Gives \( n(B) \) and \( n(A \cap B) \). Missing \( n(A) \). Not sufficient.
Combined: \( 10 + 15 – 5 = 20 \). Sufficient.
2.
Is set A a subset of set B?
(1) All elements of A are elements of B.
(2) \( A \cap B = A \).
(1) All elements of A are elements of B.
(2) \( A \cap B = A \).
Solution (D):
(1) This is the definition of a subset. Yes. Sufficient.
(2) If the intersection of A and B is A itself, it means all of A is inside B. Yes. Sufficient.
3.
How many students study both Math and Science?
(1) There are 50 students in total, and everyone studies at least one subject.
(2) 30 study Math and 35 study Science.
(1) There are 50 students in total, and everyone studies at least one subject.
(2) 30 study Math and 35 study Science.
Solution (C):
(1) Gives Total = Union = 50. Missing individual group sizes. Not sufficient.
(2) Gives group sizes. Missing total or union. Not sufficient.
Combined: \( 50 = 30 + 35 – \text{Both} \implies \text{Both} = 15 \). Sufficient.
4.
Are sets A and B disjoint (no common elements)?
(1) \( n(A) = 10 \).
(2) \( n(B) = 10 \).
(1) \( n(A) = 10 \).
(2) \( n(B) = 10 \).
Solution (E):
Knowing the size of the sets tells us nothing about their elements. They could be identical (overlap 10) or completely different (overlap 0). Not sufficient.
5.
How many people like ONLY Tea?
(1) 40 people like Tea.
(2) 15 people like both Tea and Coffee.
(1) 40 people like Tea.
(2) 15 people like both Tea and Coffee.
Solution (C):
Only Tea = Total Tea – (Tea \(\cap\) Coffee).
(1) Gives Total Tea. Not sufficient.
(2) Gives Intersection. Not sufficient.
Combined: \( 40 – 15 = 25 \). Sufficient.
6.
Is \( x \) a member of Set A?
(1) Set A contains all even integers.
(2) \( x = 12 \).
(1) Set A contains all even integers.
(2) \( x = 12 \).
Solution (C):
(1) Defines Set A, but we don’t know what \(x\) is. Not sufficient.
(2) Gives \(x\), but we don’t know what Set A is. Not sufficient.
Combined: A is set of even integers. \(x\) is 12 (which is even). So Yes. Sufficient.
7.
What is the total number of people surveyed?
(1) 20 people like A, 30 people like B, 5 like both, and 10 like neither.
(2) 50% of the people like B.
(1) 20 people like A, 30 people like B, 5 like both, and 10 like neither.
(2) 50% of the people like B.
Solution (A):
(1) Union = \(20 + 30 – 5 = 45\). Total = Union + Neither = \(45 + 10 = 55\). Sufficient.
(2) Gives percentage, but no absolute numbers. Not sufficient.
8.
Is \( n(A \cup B) > 50 \)?
(1) \( n(A) = 30 \).
(2) \( n(B) = 25 \).
(1) \( n(A) = 30 \).
(2) \( n(B) = 25 \).
Solution (E):
\( n(A \cup B) = 30 + 25 – \text{Intersection} = 55 – \text{Int} \).
We don’t know the Intersection.
If Int = 0, Union = 55 (Yes).
If Int = 20, Union = 35 (No).
Not sufficient.
9.
Are there any elements in Set A that are NOT in Set B?
(1) Set B has more elements than Set A.
(2) The intersection of A and B is equal to Set A.
(1) Set B has more elements than Set A.
(2) The intersection of A and B is equal to Set A.
Solution (B):
Question asks if \( A – B \) is empty or not.
(1) \( n(B) > n(A) \). Doesn’t mean A is inside B. They could be disjoint. Not sufficient.
(2) \( A \cap B = A \). This means all of A is inside B. Thus, there are NO elements in A that are not in B. (Answer: No). Sufficient.
10.
What percentage of the group likes both Pizza and Burger?
(1) 60% like Pizza, 50% like Burger, and 100% like at least one.
(2) 10% like neither, 60% like Pizza, and 50% like Burger.
(1) 60% like Pizza, 50% like Burger, and 100% like at least one.
(2) 10% like neither, 60% like Pizza, and 50% like Burger.
Solution (D):
(1) Union = 100%. \( 100 = 60 + 50 – \text{Both} \implies \text{Both} = 10\% \). Sufficient.
(2) Neither = 10%. Union = 90%. \( 90 = 60 + 50 – \text{Both} \implies \text{Both} = 20\% \). Sufficient.
(Note: The data in (1) and (2) contradict each other—10% vs 20%—but in DS, we evaluate each statement’s sufficiency independently. Both are sufficient).
Score: 0 / 0