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. 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…
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. If statement P is False and statement Q is True, is the implication “P → Q” True or False in classical logic?
“P → Q” · Deductive & contrapositive…
4. 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…
5. 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…
6. 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…
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 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…
9. 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…
10. 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…
11. Given “If A → B” and “If B → C”, what follows if A is true: “C is true” or “C is false”?
“If A → B” · “If B → C” · Deductive & contrapositive…
12. 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…

12 questions · Seed 104733 · 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. 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.
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. If statement P is False and statement Q is True, is the implication “P → Q” True or False in classical logic?
True
A conditional P → Q is only false when P is true and Q is false; with a false hypothesis it is vacuously true.
4. 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.
5. 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)).
6. 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”).
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 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.
9. 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.
10. 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.
11. Given “If A → B” and “If B → C”, what follows if A is true: “C is true” or “C is false”?
C is true
By hypothetical syllogism (transitivity of implication), A → B and B → C together imply A → C.
12. 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.

12 questions · Seed 104733 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Deductive & contrapositive logic

Take your time. Show your thinking.

1. 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…
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. If statement P is False and statement Q is True, is the implication “P → Q” True or False in classical logic?
“P → Q” · Deductive & contrapositive…
4. 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…
5. 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…
6. 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…
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 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…
9. 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…
10. 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…
11. Given “If A → B” and “If B → C”, what follows if A is true: “C is true” or “C is false”?
“If A → B” · “If B → C” · Deductive & contrapositive…
12. 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…

12 questions · Seed 104733 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Deductive & contrapositive logic

Answers and reasoning. Review before teaching.

1. 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.
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. If statement P is False and statement Q is True, is the implication “P → Q” True or False in classical logic?
True
A conditional P → Q is only false when P is true and Q is false; with a false hypothesis it is vacuously true.
4. 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.
5. 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)).
6. 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”).
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 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.
9. 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.
10. 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.
11. Given “If A → B” and “If B → C”, what follows if A is true: “C is true” or “C is false”?
C is true
By hypothetical syllogism (transitivity of implication), A → B and B → C together imply A → C.
12. 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.

12 questions · Seed 104733 · Age guidance, not curriculum certification