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. 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…
3. 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…
4. 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…
5. 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…
6. 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…
7. 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…
8. 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…
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 logical negation of “Some integers are negative”: “No integers are negative” or “Some integers are positive”?
What is the logical negati… · Deductive & contrapositive…
11. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
“Having four sides” · 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 104748 · 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. 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.
3. 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.
4. 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)).
5. 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.
6. 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.
7. 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.
8. 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.
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 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”).
11. 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. 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 104748 · 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. 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…
3. 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…
4. 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…
5. 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…
6. 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…
7. 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…
8. 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…
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 logical negation of “Some integers are negative”: “No integers are negative” or “Some integers are positive”?
What is the logical negati… · Deductive & contrapositive…
11. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
“Having four sides” · 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 104748 · 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. 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.
3. 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.
4. 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)).
5. 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.
6. 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.
7. 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.
8. 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.
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 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”).
11. 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. 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 104748 · Age guidance, not curriculum certification