Mathematical Reasoning & Logical Thinking Flashcards

Which of the following is the contrapositive of the statement 'If n is divisible by 6, then n is divisible by 2'?

Answer: If n is not divisible by 2, then n is not divisible by 6

The contrapositive of 'If P then Q' is 'If not Q then not P', so 'If n is not divisible by 2, then n is not divisible by 6'.

A sequence is defined by a₁ = 3 and aₙ₊₁ = 2aₙ − 1. What is a₄?

Answer: 17

a₂ = 2(3)−1 = 5, a₃ = 2(5)−1 = 9, a₄ = 2(9)−1 = 17.

For integers m and n, which statement is always true if m and n are both odd?

Answer: m − n is even

Odd minus odd equals even, since (2j+1)−(2k+1) = 2(j−k), which is always even.

How many integers between 1 and 100 inclusive are divisible by 3 but not by 9?

Answer: 22

Divisible by 3: floor(100/3)=33; divisible by 9: floor(100/9)=11; answer = 33−11 = 22.

If p → q and q → r are both true, which conclusion must follow by the chain rule?

Answer: p → r

Hypothetical syllogism (chain rule) states: if p→q and q→r, then p→r.

A fair six-sided die is rolled twice. What is the probability that the product of the two results is even?

Answer: 3/4

P(product odd) = P(both odd) = (1/2)² = 1/4, so P(product even) = 1 − 1/4 = 3/4.

Which of the following is a necessary condition for a natural number n to be prime (other than n = 2)?

Answer: n is odd

Every prime greater than 2 must be odd; being odd is therefore a necessary (but not sufficient) condition.