Online Test — Mathematical Reasoning
10 Questions • 20 min • Chapter MCQ
20:00
Question 1 of 10
easy
Which of the following is a mathematically acceptable statement?
Please sit down.
How wonderful!
The number 12 is divisible by 4.
Where are you going?
Explanation: A statement is a sentence that is true or false. "12 is divisible by 4" is true (a definite truth value). The others are a command, an exclamation and a question.
Question 2 of 10
medium
The negation of the statement "All birds can fly" is:
No bird can fly.
All birds cannot fly.
There exists a bird that cannot fly.
Some birds can fly.
Explanation: Negating a "for all" statement gives "there exists … not": there is at least one bird that cannot fly.
Question 3 of 10
easy
The compound statement "p and q" (p ∧ q) is true when:
at least one of p, q is true
both p and q are true
both p and q are false
exactly one is true
Explanation: A conjunction is true only when both component statements are true.
Question 4 of 10
medium
The conditional "if p then q" is FALSE only in which case?
p true, q true
p true, q false
p false, q true
p false, q false
Explanation: A conditional fails only when the hypothesis is true but the conclusion is false.
Question 5 of 10
medium
The contrapositive of "If a number is divisible by 6, then it is divisible by 3" is:
If a number is divisible by 3, then it is divisible by 6.
If a number is not divisible by 3, then it is not divisible by 6.
If a number is not divisible by 6, then it is not divisible by 3.
If a number is divisible by 6, then it is not divisible by 3.
Explanation: The contrapositive of p ⇒ q is (~q) ⇒ (~p): "not divisible by 3 ⇒ not divisible by 6".
Question 6 of 10
easy
Which word is the logical connective in "It is hot or it is humid"?
it
is
or
hot
Explanation: "Or" joins the two component statements as a disjunction.
Question 7 of 10
medium
A single counter-example is sufficient to:
prove a "for all" statement
disprove a "for all" statement
prove an "if and only if" statement
prove a conditional
Explanation: One exception destroys a universal ("for all") claim, but cannot prove one.
Question 8 of 10
medium
The statement "p if and only if q" is equivalent to:
p ⇒ q only
q ⇒ p only
(p ⇒ q) and (q ⇒ p)
p or q
Explanation: A biconditional is the conjunction of a conditional and its converse.
Question 9 of 10
hard
In "p ⇒ q", the statement "q is necessary for p" means:
q can hold without p
p cannot be true unless q is true
p is required for q
p and q are the same
Explanation: In a conditional, the conclusion q is necessary: p cannot be true while q is false.
Question 10 of 10
medium
The proof that √2 is irrational typically uses the method of:
direct proof
contradiction
counter-example
converse
Explanation: Assuming √2 is rational leads to a contradiction (lowest-terms fraction with both terms even), so the method is proof by contradiction.