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. If “P ∧ Q” (P and Q) is True, what must be the truth value of P: True or False?
“P ∧ Q” · Deductive & contrapositive…
3. 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…
4. 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…
5. 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…
6. 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…
7. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
“Having four sides” · Deductive & contrapositive…
8. 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…
9. 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…
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. 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…
12. 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…

12 questions · Seed 104761 · 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. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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).
8. 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.
9. 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.
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. 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”).
12. 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.

12 questions · Seed 104761 · 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. If “P ∧ Q” (P and Q) is True, what must be the truth value of P: True or False?
“P ∧ Q” · Deductive & contrapositive…
3. 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…
4. 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…
5. 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…
6. 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…
7. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
“Having four sides” · Deductive & contrapositive…
8. 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…
9. 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…
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. 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…
12. 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…

12 questions · Seed 104761 · 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. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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).
8. 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.
9. 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.
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. 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”).
12. 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.

12 questions · Seed 104761 · Age guidance, not curriculum certification