Mathematical Reasoning & Logical Thinking Flashcards

Which of the following arguments is logically valid?

Answer: P→Q, Q is false, therefore P is false

Modus tollens: if P→Q and ¬Q, then ¬P; this is the only valid argument of the four.

A bag contains 4 red and 6 blue balls. Two balls are drawn without replacement. What is the probability both are red?

Answer: 2/15

P = (4/10) × (3/9) = 12/90 = 2/15.

The proposition '∃x ∈ ℤ such that x² = 2' is:

Answer: False, since √2 is irrational and not an integer

√2 is irrational and not an integer, so no integer x satisfies x²=2, making the statement false.

Using the pigeonhole principle, what is the minimum number of students needed to guarantee that at least 3 share the same birth month?

Answer: 25

With 12 months, distributing 24 students allows 2 per month; the 25th guarantees a third in some month.

Which of the following is the negation of 'All prime numbers greater than 2 are odd'?

Answer: There exists a prime number greater than 2 that is not odd

The negation of '∀x, P(x)' is '∃x, ¬P(x)', i.e., there exists a prime greater than 2 that is not odd.

If f(x) = 2x + 1 and g(x) = x², what is g(f(3))?

Answer: 49

f(3) = 2(3)+1 = 7; g(7) = 7² = 49.

A truth table for P XOR Q (exclusive or) shows P XOR Q is true when:

Answer: Exactly one of P and Q is true

XOR is true precisely when the operands differ in truth value — one true and one false.