Mathematical Reasoning & Logical Thinking Flashcards

Let P = 'x > 0' and Q = 'x² > 0'. Which relationship correctly describes P and Q for real x?

Answer: P is sufficient but not necessary for Q

P→Q (if x>0 then x²>0) but Q does not imply P since x=−2 gives x²=4>0; so P is sufficient but not necessary.

A function f satisfies f(x + y) = f(x) + f(y) for all real x, y, and f(1) = 3. What is f(7)?

Answer: 21

By the Cauchy functional equation with f(1)=3, we get f(n)=3n for integers, so f(7)=21.

In a group of 30 students, 18 study Maths, 15 study Physics, and 5 study neither. How many study both?

Answer: 8

By inclusion-exclusion: |M∪P| = 30−5 = 25, so |M∩P| = 18+15−25 = 8.

Which of the following correctly negates the statement '∀x ∈ ℝ, x² ≥ 0'?

Answer: ∃x ∈ ℝ such that x² < 0

Negating a universal statement gives an existential statement with the negated predicate: ∃x such that x² < 0.

If the sum of three consecutive even integers is 78, what is the largest of the three?

Answer: 28

Let the integers be n, n+2, n+4; then 3n+6=78, so n=24 and the largest is 28.

A statement and its converse are both true. What can be concluded?

Answer: The statement and its converse form a biconditional

When both 'P→Q' and 'Q→P' are true, together they constitute the biconditional 'P↔Q'.

How many ways can 3 distinct books be arranged on a shelf from a collection of 7 distinct books?

Answer: 210

This is a permutation: P(7,3) = 7 × 6 × 5 = 210.