ReasoningAges 16-18

Deductive & contrapositive logic

Evaluate conditional statements, contrapositives and valid deductive inferences. Review suitability for your learners before teaching.

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Worksheet StudioPractice pack · Ages 16-18

Deductive & contrapositive logic

Take your time. Show your thinking.

1. If statement P is False and statement Q is True, is the implication “P → Q” True or False in classical logic?
“P → Q” · Deductive & contrapositive…
2. If “P ∨ Q” (P or Q) is False, what must be the truth value of both P and Q: Both False, or Both True?
“P ∨ Q” · Deductive & contrapositive…
3. Given “All prime numbers greater than 2 are odd” and “p = 17 is a prime greater than 2”, is 17 odd (Yes / No)?
2 · 17 · 2 · 17 · Deductive & contrapositive…
4. If “P ∧ Q” (P and Q) is True, what must be the truth value of P: True or False?
“P ∧ Q” · Deductive & contrapositive…
5. What single counterexample disproves the universal claim “All prime numbers are odd”?
What single counterexample… · Deductive & contrapositive…
6. Given “Either the fuse blew OR the switch is off” and “The switch is NOT off”, did the fuse blow (Yes / No)?
Given “Either the fuse ble… · Deductive & contrapositive…
7. Given “If a figure is a square, then it has four equal sides” and “Figure X does NOT have four equal sides”, can X be a square (Yes / No)?
Given “If a figure is a sq… · Deductive & contrapositive…
8. Given “If the solution turns blue litmus red, it is acidic” and “The litmus did NOT turn red”, is the solution acidic (Yes / No)?
Given “If the solution tur… · Deductive & contrapositive…
9. Given “If a triangle is equilateral, all its interior angles are 60°” and “Triangle T has an angle of 90°”, can T be equilateral (Yes / No)?
60° · 90° · Deductive & contrapositive…
10. Does “If a metal is iron, it is magnetic” logically imply “If a metal is magnetic, it must be iron” (Yes / No)?
Does “If a metal is iron, … · Deductive & contrapositive…
11. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
“1” · “3” · Deductive & contrapositive…
12. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
4 · Deductive & contrapositive…

12 questions · Seed 104729 · Age guidance, not curriculum certification

View the answer key

Worksheet StudioAnswer key · Ages 16-18

Deductive & contrapositive logic

Answers and reasoning. Review before teaching.

1. If statement P is False and statement Q is True, is the implication “P → Q” True or False in classical logic?
True
A conditional P → Q is only false when P is true and Q is false; with a false hypothesis it is vacuously true.
2. If “P ∨ Q” (P or Q) is False, what must be the truth value of both P and Q: Both False, or Both True?
Both False
An inclusive disjunction P ∨ Q is false only when both P and Q are false.
3. Given “All prime numbers greater than 2 are odd” and “p = 17 is a prime greater than 2”, is 17 odd (Yes / No)?
Yes
Universal instantiation applies the rule to p = 17, so 17 is odd.
4. If “P ∧ Q” (P and Q) is True, what must be the truth value of P: True or False?
True
A conjunction P ∧ Q is true only when both P is true and Q is true.
5. What single counterexample disproves the universal claim “All prime numbers are odd”?
2
2 is a prime number (divisible only by 1 and 2) and is even, disproving the universal claim.
6. Given “Either the fuse blew OR the switch is off” and “The switch is NOT off”, did the fuse blow (Yes / No)?
Yes
By disjunctive syllogism (P or Q; not Q; therefore P), the fuse blew.
7. Given “If a figure is a square, then it has four equal sides” and “Figure X does NOT have four equal sides”, can X be a square (Yes / No)?
No
By Modus Tollens (if P then Q; not Q; therefore not P), X cannot be a square.
8. Given “If the solution turns blue litmus red, it is acidic” and “The litmus did NOT turn red”, is the solution acidic (Yes / No)?
No
By Modus Tollens, since the consequent is false, the antecedent must be false.
9. Given “If a triangle is equilateral, all its interior angles are 60°” and “Triangle T has an angle of 90°”, can T be equilateral (Yes / No)?
No
By Modus Tollens, since not all angles of T are 60°, T cannot be equilateral.
10. Does “If a metal is iron, it is magnetic” logically imply “If a metal is magnetic, it must be iron” (Yes / No)?
No
Assuming the converse is a fallacy; nickel and cobalt are also magnetic metals.
11. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
1
For n = 1, 1² = 1, which is not greater than 1 (n = 0 also works).
12. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
Sufficient
Every multiple of 4 is guaranteed to be even (sufficient), though an even number like 6 need not be a multiple of 4.

12 questions · Seed 104729 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Deductive & contrapositive logic

Take your time. Show your thinking.

1. If statement P is False and statement Q is True, is the implication “P → Q” True or False in classical logic?
“P → Q” · Deductive & contrapositive…
2. If “P ∨ Q” (P or Q) is False, what must be the truth value of both P and Q: Both False, or Both True?
“P ∨ Q” · Deductive & contrapositive…
3. Given “All prime numbers greater than 2 are odd” and “p = 17 is a prime greater than 2”, is 17 odd (Yes / No)?
2 · 17 · 2 · 17 · Deductive & contrapositive…
4. If “P ∧ Q” (P and Q) is True, what must be the truth value of P: True or False?
“P ∧ Q” · Deductive & contrapositive…
5. What single counterexample disproves the universal claim “All prime numbers are odd”?
What single counterexample… · Deductive & contrapositive…
6. Given “Either the fuse blew OR the switch is off” and “The switch is NOT off”, did the fuse blow (Yes / No)?
Given “Either the fuse ble… · Deductive & contrapositive…
7. Given “If a figure is a square, then it has four equal sides” and “Figure X does NOT have four equal sides”, can X be a square (Yes / No)?
Given “If a figure is a sq… · Deductive & contrapositive…
8. Given “If the solution turns blue litmus red, it is acidic” and “The litmus did NOT turn red”, is the solution acidic (Yes / No)?
Given “If the solution tur… · Deductive & contrapositive…
9. Given “If a triangle is equilateral, all its interior angles are 60°” and “Triangle T has an angle of 90°”, can T be equilateral (Yes / No)?
60° · 90° · Deductive & contrapositive…
10. Does “If a metal is iron, it is magnetic” logically imply “If a metal is magnetic, it must be iron” (Yes / No)?
Does “If a metal is iron, … · Deductive & contrapositive…
11. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
“1” · “3” · Deductive & contrapositive…
12. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
4 · Deductive & contrapositive…

12 questions · Seed 104729 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Deductive & contrapositive logic

Answers and reasoning. Review before teaching.

1. If statement P is False and statement Q is True, is the implication “P → Q” True or False in classical logic?
True
A conditional P → Q is only false when P is true and Q is false; with a false hypothesis it is vacuously true.
2. If “P ∨ Q” (P or Q) is False, what must be the truth value of both P and Q: Both False, or Both True?
Both False
An inclusive disjunction P ∨ Q is false only when both P and Q are false.
3. Given “All prime numbers greater than 2 are odd” and “p = 17 is a prime greater than 2”, is 17 odd (Yes / No)?
Yes
Universal instantiation applies the rule to p = 17, so 17 is odd.
4. If “P ∧ Q” (P and Q) is True, what must be the truth value of P: True or False?
True
A conjunction P ∧ Q is true only when both P is true and Q is true.
5. What single counterexample disproves the universal claim “All prime numbers are odd”?
2
2 is a prime number (divisible only by 1 and 2) and is even, disproving the universal claim.
6. Given “Either the fuse blew OR the switch is off” and “The switch is NOT off”, did the fuse blow (Yes / No)?
Yes
By disjunctive syllogism (P or Q; not Q; therefore P), the fuse blew.
7. Given “If a figure is a square, then it has four equal sides” and “Figure X does NOT have four equal sides”, can X be a square (Yes / No)?
No
By Modus Tollens (if P then Q; not Q; therefore not P), X cannot be a square.
8. Given “If the solution turns blue litmus red, it is acidic” and “The litmus did NOT turn red”, is the solution acidic (Yes / No)?
No
By Modus Tollens, since the consequent is false, the antecedent must be false.
9. Given “If a triangle is equilateral, all its interior angles are 60°” and “Triangle T has an angle of 90°”, can T be equilateral (Yes / No)?
No
By Modus Tollens, since not all angles of T are 60°, T cannot be equilateral.
10. Does “If a metal is iron, it is magnetic” logically imply “If a metal is magnetic, it must be iron” (Yes / No)?
No
Assuming the converse is a fallacy; nickel and cobalt are also magnetic metals.
11. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
1
For n = 1, 1² = 1, which is not greater than 1 (n = 0 also works).
12. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
Sufficient
Every multiple of 4 is guaranteed to be even (sufficient), though an even number like 6 need not be a multiple of 4.

12 questions · Seed 104729 · Age guidance, not curriculum certification