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. 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…
2. 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…
3. What is the logical negation of “Some integers are negative”: “No integers are negative” or “Some integers are positive”?
What is the logical negati… · Deductive & contrapositive…
4. 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…
5. What is the logical negation of “All birds can fly”: “At least one bird cannot fly” or “No birds can fly”?
“All birds can fly” · “No bird · Deductive & contrapositive…
6. What single counterexample disproves the universal claim “All prime numbers are odd”?
What single counterexample… · Deductive & contrapositive…
7. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
4 · Deductive & contrapositive…
8. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
“1” · “3” · Deductive & contrapositive…
9. 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…
10. What is the contrapositive of “If it rains, the pitch is wet”: “If the pitch is not wet, it did not rain” or “If the pitch is wet, it rained”?
What is the contrapositive… · Deductive & contrapositive…
11. In a biconditional statement “P if and only if Q”, when is the statement True: when P and Q have the same truth value, or when P and Q have opposite truth values?
“P if and only if Q” · Deductive & contrapositive…
12. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
“Having four sides” · Deductive & contrapositive…

12 questions · Seed 104731 · 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. 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.
2. 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.
3. What is the logical negation of “Some integers are negative”: “No integers are negative” or “Some integers are positive”?
No integers are negative
The negation of ∃x P(x) is ∀x ¬P(x) (“No integers are negative”).
4. 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.
5. What is the logical negation of “All birds can fly”: “At least one bird cannot fly” or “No birds can fly”?
At least one bird cannot fly
The negation of a universal statement (∀x P(x)) is an existential statement (∃x ¬P(x)).
6. 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.
7. 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.
8. 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).
9. 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.
10. What is the contrapositive of “If it rains, the pitch is wet”: “If the pitch is not wet, it did not rain” or “If the pitch is wet, it rained”?
If the pitch is not wet, it did not rain
The contrapositive of “If P then Q” is “If not Q then not P”, which is logically equivalent.
11. In a biconditional statement “P if and only if Q”, when is the statement True: when P and Q have the same truth value, or when P and Q have opposite truth values?
When P and Q have the same truth value
P ↔ Q is true when both P and Q are true, or when both P and Q are false.
12. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
Necessary
A square must have four sides (necessary), but a rectangle or rhombus also has four sides (so it is not sufficient).

12 questions · Seed 104731 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Deductive & contrapositive logic

Take your time. Show your thinking.

1. 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…
2. 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…
3. What is the logical negation of “Some integers are negative”: “No integers are negative” or “Some integers are positive”?
What is the logical negati… · Deductive & contrapositive…
4. 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…
5. What is the logical negation of “All birds can fly”: “At least one bird cannot fly” or “No birds can fly”?
“All birds can fly” · “No bird · Deductive & contrapositive…
6. What single counterexample disproves the universal claim “All prime numbers are odd”?
What single counterexample… · Deductive & contrapositive…
7. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
4 · Deductive & contrapositive…
8. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
“1” · “3” · Deductive & contrapositive…
9. 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…
10. What is the contrapositive of “If it rains, the pitch is wet”: “If the pitch is not wet, it did not rain” or “If the pitch is wet, it rained”?
What is the contrapositive… · Deductive & contrapositive…
11. In a biconditional statement “P if and only if Q”, when is the statement True: when P and Q have the same truth value, or when P and Q have opposite truth values?
“P if and only if Q” · Deductive & contrapositive…
12. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
“Having four sides” · Deductive & contrapositive…

12 questions · Seed 104731 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Deductive & contrapositive logic

Answers and reasoning. Review before teaching.

1. 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.
2. 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.
3. What is the logical negation of “Some integers are negative”: “No integers are negative” or “Some integers are positive”?
No integers are negative
The negation of ∃x P(x) is ∀x ¬P(x) (“No integers are negative”).
4. 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.
5. What is the logical negation of “All birds can fly”: “At least one bird cannot fly” or “No birds can fly”?
At least one bird cannot fly
The negation of a universal statement (∀x P(x)) is an existential statement (∃x ¬P(x)).
6. 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.
7. 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.
8. 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).
9. 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.
10. What is the contrapositive of “If it rains, the pitch is wet”: “If the pitch is not wet, it did not rain” or “If the pitch is wet, it rained”?
If the pitch is not wet, it did not rain
The contrapositive of “If P then Q” is “If not Q then not P”, which is logically equivalent.
11. In a biconditional statement “P if and only if Q”, when is the statement True: when P and Q have the same truth value, or when P and Q have opposite truth values?
When P and Q have the same truth value
P ↔ Q is true when both P and Q are true, or when both P and Q are false.
12. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
Necessary
A square must have four sides (necessary), but a rectangle or rhombus also has four sides (so it is not sufficient).

12 questions · Seed 104731 · Age guidance, not curriculum certification