Mathematical Reasoning & Logical Thinking Flashcards

Which of the following is an example of a proof by contradiction?

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.

Given that log₂(x) + log₂(x−2) = 3, what is x?

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).

How many subsets does a set with 4 elements have?

Answer: 16

A set with n elements has 2ⁿ subsets; for n=4, that is 2⁴=16.

Two events A and B are mutually exclusive. If P(A)=0.3 and P(B)=0.4, what is P(A or B)?

Answer: 0.70

For mutually exclusive events, P(A∪B)=P(A)+P(B)=0.3+0.4=0.7.

What is the smallest positive integer n such that n! is divisible by 10⁶?

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.

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}?

Answer: {3,5}

A' = {1,3,5,7}; intersecting with {3,4,5,6} gives {3,5}.

Which of the following statements about proof by induction is 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.