TMUA Mathematical Reasoning & Logical Thinking 4 — Questions and Answers
Question 1: Which of the following is an example of a proof by contradiction?
- Assuming P is true and deriving Q directly
- Assuming ¬P and showing this leads to a logical impossibility (Correct answer)
- Proving P true for a base case and an inductive step
- Testing P for several specific values of n
Correct answer: Assuming ¬P and showing this leads to a logical impossibility
Proof by contradiction assumes the negation of the claim and derives a contradiction, establishing the original claim.
Question 2: Given that log₂(x) + log₂(x−2) = 3, what is x?
- 4 (Correct answer)
- 2
- 6
- 8
Correct answer: 4
log₂(x(x−2))=3 gives x(x−2)=8, so x²−2x−8=0, factoring to (x−4)(x+2)=0; x=4 (rejecting x=−2 as x>2).
Question 3: How many subsets does a set with 4 elements have?
- 8
- 12
- 16 (Correct answer)
- 24
Correct answer: 16
A set with n elements has 2ⁿ subsets; for n=4, that is 2⁴=16.
Question 4: Two events A and B are mutually exclusive. If P(A)=0.3 and P(B)=0.4, what is P(A or B)?
- 0.12
- 0.58
- 0.70 (Correct answer)
- 0.7
Correct answer: 0.70
For mutually exclusive events, P(A∪B)=P(A)+P(B)=0.3+0.4=0.7.
Question 5: What is the smallest positive integer n such that n! is divisible by 10⁶?
- 20
- 22
- 24
- 25 (Correct answer)
Correct answer: 25
10⁶=2⁶×5⁶; the number of factors of 5 in n! determines the constraint. 25! contributes floor(25/5)+floor(25/25)=5+1=6 factors of 5, so n=25.
Question 6: If the universal set U = {1,2,3,4,5,6,7,8} and A = {2,4,6,8}, what is A' ∩ {3,4,5,6}?
- {3,5} (Correct answer)
- {4,6}
- {3,4,5,6}
- {2,4,6,8}
Correct answer: {3,5}
A' = {1,3,5,7}; intersecting with {3,4,5,6} gives {3,5}.
Question 7: Which of the following statements about proof by induction is correct?
- You only need to verify the base case
- The inductive step proves the statement true for all n directly
- You assume the statement holds for n=k and prove it for n=k+1 (Correct answer)
- The base case must always be n=0
Correct answer: You assume the statement holds for n=k and prove it for n=k+1
The inductive step assumes the statement is true for some arbitrary k (inductive hypothesis) and proves it for k+1.
TMUA Mathematical Reasoning & Logical Thinking 4
Which of the following is an example of a proof by contradiction?