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. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
4 · Deductive & contrapositive…
2. By De Morgan’s laws, the negation of “Either X or Y is true” is: “Neither X nor Y is true” or “Only X is false”?
“Only X is false” · Deductive & contrapositive…
3. Given “If a number is divisible by 6, then it is even” and “N is divisible by 6”, is N even (Yes / No)?
6 · 6 · Deductive & contrapositive…
4. 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…
5. 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…
6. Given “Every square is a rectangle” and “Every rectangle is a parallelogram”, does it follow that “Every square is a parallelogram” (Yes / No)?
Given “Every square is a r… · Deductive & contrapositive…
7. If “P ∧ Q” (P and Q) is True, what must be the truth value of P: True or False?
“P ∧ Q” · Deductive & contrapositive…
8. Given “No reptiles have fur” and “All lemurs have fur”, can any lemur be a reptile (Yes / No)?
Given “No reptiles have fu… · Deductive & contrapositive…
9. What single counterexample disproves the universal claim “All prime numbers are odd”?
What single counterexample… · Deductive & contrapositive…
10. By De Morgan’s laws, the negation of “Both X and Y are true” is: “At least one of X or Y is false” or “Both X and Y are false”?
By De Morgan’s laws, the n… · Deductive & contrapositive…
11. 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…
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 104738 · 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. 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.
2. By De Morgan’s laws, the negation of “Either X or Y is true” is: “Neither X nor Y is true” or “Only X is false”?
Neither X nor Y is true
¬(X ∨ Y) is logically equivalent to (¬X ∧ ¬Y): both X and Y are false.
3. Given “If a number is divisible by 6, then it is even” and “N is divisible by 6”, is N even (Yes / No)?
Yes
By Modus Ponens (if P then Q; P is true; therefore Q is true), N must be even.
4. 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.
5. 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.
6. Given “Every square is a rectangle” and “Every rectangle is a parallelogram”, does it follow that “Every square is a parallelogram” (Yes / No)?
Yes
Subset inclusion is transitive: Squares ⊆ Rectangles ⊆ Parallelograms.
7. 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.
8. Given “No reptiles have fur” and “All lemurs have fur”, can any lemur be a reptile (Yes / No)?
No
Since reptiles have no fur and lemurs have fur, the two sets are disjoint.
9. 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.
10. By De Morgan’s laws, the negation of “Both X and Y are true” is: “At least one of X or Y is false” or “Both X and Y are false”?
At least one of X or Y is false
¬(X ∧ Y) is logically equivalent to (¬X ∨ ¬Y): at least one is false.
11. 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.
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 104738 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Deductive & contrapositive logic

Take your time. Show your thinking.

1. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
4 · Deductive & contrapositive…
2. By De Morgan’s laws, the negation of “Either X or Y is true” is: “Neither X nor Y is true” or “Only X is false”?
“Only X is false” · Deductive & contrapositive…
3. Given “If a number is divisible by 6, then it is even” and “N is divisible by 6”, is N even (Yes / No)?
6 · 6 · Deductive & contrapositive…
4. 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…
5. 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…
6. Given “Every square is a rectangle” and “Every rectangle is a parallelogram”, does it follow that “Every square is a parallelogram” (Yes / No)?
Given “Every square is a r… · Deductive & contrapositive…
7. If “P ∧ Q” (P and Q) is True, what must be the truth value of P: True or False?
“P ∧ Q” · Deductive & contrapositive…
8. Given “No reptiles have fur” and “All lemurs have fur”, can any lemur be a reptile (Yes / No)?
Given “No reptiles have fu… · Deductive & contrapositive…
9. What single counterexample disproves the universal claim “All prime numbers are odd”?
What single counterexample… · Deductive & contrapositive…
10. By De Morgan’s laws, the negation of “Both X and Y are true” is: “At least one of X or Y is false” or “Both X and Y are false”?
By De Morgan’s laws, the n… · Deductive & contrapositive…
11. 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…
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 104738 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Deductive & contrapositive logic

Answers and reasoning. Review before teaching.

1. 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.
2. By De Morgan’s laws, the negation of “Either X or Y is true” is: “Neither X nor Y is true” or “Only X is false”?
Neither X nor Y is true
¬(X ∨ Y) is logically equivalent to (¬X ∧ ¬Y): both X and Y are false.
3. Given “If a number is divisible by 6, then it is even” and “N is divisible by 6”, is N even (Yes / No)?
Yes
By Modus Ponens (if P then Q; P is true; therefore Q is true), N must be even.
4. 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.
5. 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.
6. Given “Every square is a rectangle” and “Every rectangle is a parallelogram”, does it follow that “Every square is a parallelogram” (Yes / No)?
Yes
Subset inclusion is transitive: Squares ⊆ Rectangles ⊆ Parallelograms.
7. 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.
8. Given “No reptiles have fur” and “All lemurs have fur”, can any lemur be a reptile (Yes / No)?
No
Since reptiles have no fur and lemurs have fur, the two sets are disjoint.
9. 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.
10. By De Morgan’s laws, the negation of “Both X and Y are true” is: “At least one of X or Y is false” or “Both X and Y are false”?
At least one of X or Y is false
¬(X ∧ Y) is logically equivalent to (¬X ∨ ¬Y): at least one is false.
11. 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.
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 104738 · Age guidance, not curriculum certification