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. 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…
2. 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…
3. What single counterexample disproves the universal claim “All prime numbers are odd”?
What single counterexample… · Deductive & contrapositive…
4. 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…
5. Does “If n is divisible by 10, then n is divisible by 5” have a true contrapositive: “If n is not divisible by 5, then n is not divisible by 10” (Yes / No)?
10 · 5 · 5 · 10 · Deductive & contrapositive…
6. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
4 · Deductive & contrapositive…
7. 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…
8. 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…
9. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
“1” · “3” · Deductive & contrapositive…
10. 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…
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. 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…

12 questions · Seed 104735 · 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. 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.
2. 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.
3. 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.
4. 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.
5. Does “If n is divisible by 10, then n is divisible by 5” have a true contrapositive: “If n is not divisible by 5, then n is not divisible by 10” (Yes / No)?
Yes
A conditional and its contrapositive are always logically equivalent.
6. 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.
7. 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.
8. 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.
9. 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).
10. 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.
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. 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.

12 questions · Seed 104735 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Deductive & contrapositive logic

Take your time. Show your thinking.

1. 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…
2. 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…
3. What single counterexample disproves the universal claim “All prime numbers are odd”?
What single counterexample… · Deductive & contrapositive…
4. 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…
5. Does “If n is divisible by 10, then n is divisible by 5” have a true contrapositive: “If n is not divisible by 5, then n is not divisible by 10” (Yes / No)?
10 · 5 · 5 · 10 · Deductive & contrapositive…
6. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
4 · Deductive & contrapositive…
7. 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…
8. 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…
9. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
“1” · “3” · Deductive & contrapositive…
10. 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…
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. 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…

12 questions · Seed 104735 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Deductive & contrapositive logic

Answers and reasoning. Review before teaching.

1. 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.
2. 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.
3. 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.
4. 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.
5. Does “If n is divisible by 10, then n is divisible by 5” have a true contrapositive: “If n is not divisible by 5, then n is not divisible by 10” (Yes / No)?
Yes
A conditional and its contrapositive are always logically equivalent.
6. 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.
7. 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.
8. 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.
9. 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).
10. 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.
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. 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.

12 questions · Seed 104735 · Age guidance, not curriculum certification