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 number is divisible by 6, then it is even” and “N is divisible by 6”, is N even (Yes / No)?
6 · 6 · Deductive & contrapositive…
2. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
“1” · “3” · Deductive & contrapositive…
3. 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…
4. 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…
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. If “P ∧ Q” (P and Q) is True, what must be the truth value of P: True or False?
“P ∧ Q” · Deductive & contrapositive…
7. 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…
8. 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…
9. 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…
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. 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…
12. 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…

12 questions · Seed 104758 · 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 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.
2. 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).
3. 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.
4. 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.
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. 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.
7. 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.
8. 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.
9. 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.
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. 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.
12. 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.

12 questions · Seed 104758 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Deductive & contrapositive logic

Take your time. Show your thinking.

1. 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…
2. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
“1” · “3” · Deductive & contrapositive…
3. 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…
4. 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…
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. If “P ∧ Q” (P and Q) is True, what must be the truth value of P: True or False?
“P ∧ Q” · Deductive & contrapositive…
7. 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…
8. 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…
9. 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…
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. 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…
12. 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…

12 questions · Seed 104758 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Deductive & contrapositive logic

Answers and reasoning. Review before teaching.

1. 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.
2. 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).
3. 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.
4. 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.
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. 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.
7. 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.
8. 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.
9. 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.
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. 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.
12. 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.

12 questions · Seed 104758 · Age guidance, not curriculum certification