TMUA Mathematical Reasoning & Logical Thinking 5

Question 1

Which of the following arguments is logically valid?

Accepted Answer

P→Q, Q is false, therefore P is false

Answer

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

Question 2

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

Accepted Answer

2/15

Answer

P = (4/10) × (3/9) = 12/90 = 2/15.

Question 3

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

Accepted Answer

False, since √2 is irrational and not an integer

Answer

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

Question 4

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

Accepted Answer

25

Answer

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

Question 5

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

Accepted Answer

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

Answer

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

(TMUA) Test of Mathematics for University Admission Practice Test