SET Practice SET – 5
Set Theory: Level 5
1.
Set A has 40 members, Set B has 60 members. If \( A \cap B \) has 25 members, what is the number of members in \( B – A \) (in B but not in A)?
Solution (B):
\( n(B – A) = n(B) – n(A \cap B) \).
\( 60 – 25 = 35 \).
2.
In a survey of 500 people: 80% have a TV, 60% have a Computer, and 50% have a Tablet. What is the **maximum possible** number of people who have **none** of the three?
Solution (D):
To maximize “None”, we must minimize the Union.
To minimize Union (A \(\cup\) B \(\cup\) C), we make the sets overlap as much as possible.
The largest set is A (80%). We can fit B (60%) completely inside A, and C (50%) completely inside A (and B).
If \( C \subset B \subset A \), then the Union is just size of A = 80%.
If Union is 80%, then None = \( 100\% – 80\% = 20\% \).
\( 20\% \text{ of } 500 = 100 \).
3.
Of 200 students, 100 take Math, 80 take History. If 40 students take neither, what is the ratio of students who take **only Math** to students who take **only History**?
Solution (C):
Union = \( 200 – 40 = 160 \).
\( 160 = 100 + 80 – \text{Both} \).
\( 160 = 180 – \text{Both} \implies \text{Both} = 20 \).
Only Math = \( 100 – 20 = 80 \).
Only History = \( 80 – 20 = 60 \).
Ratio = \( 80 : 60 = 4 : 3 \).
4.
In a certain town, 40% of people like Tea. 30% like Coffee. 10% like neither. What percentage of people like **exactly one** of the two?
Solution (E):
Wait, let’s check logic.
Union = \( 100 – 10 = 90\% \).
\( 90 = 40 + 30 – \text{Both} \).
\( 90 = 70 – \text{Both} \implies \text{Both} = -20\% \). Impossible.
This implies the problem data is invalid (40+30 < 90). The sets are disjoint and cover only 70%, so "Neither" must be at least 30%.
However, assuming this is a trick question on constraints:
Let's re-read: maybe "30% like Coffee" is independent?
No, the data 40+30=70 is less than the required Union (90).
This means the Intersection cannot exist (is negative).
Let's assume the question meant: **10% like BOTH**. Then find Neither?
Or perhaps 10% like Neither is the typo, should be 30% or more.
Let's adjust the question to valid numbers for the quiz:
**New Numbers:** 60% Tea, 50% Coffee, 10% Neither.
Union = 90.
\( 90 = 60 + 50 - \text{Both} \implies 90 = 110 - \text{Both} \implies \text{Both} = 20 \).
Only T = \( 60 - 20 = 40 \). Only C = \( 50 - 20 = 30 \).
Exactly one = \( 40 + 30 = 70\% \).
Answer E (70%) works with these numbers.
4.
In a certain town, 60% of people like Tea, 50% like Coffee, and 10% like neither. What percentage of people like **exactly one** of the two?
Solution (E):
Union = \( 100 – 10 = 90\% \).
\( 90 = 60 + 50 – \text{Both} \implies \text{Both} = 20\% \).
Only Tea = \( 60 – 20 = 40\% \).
Only Coffee = \( 50 – 20 = 30\% \).
Exactly One = \( 40 + 30 = 70\% \).
5.
Set P contains all multiples of 4 less than 50. Set Q contains all multiples of 6 less than 50. How many elements are in \( P \cap Q \)?
Solution (A):
\( P \cap Q \) contains multiples of LCM(4, 6) = 12.
Multiples of 12 less than 50: 12, 24, 36, 48.
There are 4 elements.
Score: 0 / 0