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. 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…
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. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
“Having four sides” · Deductive & contrapositive…
5. 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…
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. 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…
8. What single counterexample disproves the universal claim “All prime numbers are odd”?
What single counterexample… · Deductive & contrapositive…
9. 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…
10. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
4 · 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 “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 104753 · 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. 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.
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. 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).
5. 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.
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. 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.
8. 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.
9. 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.
10. 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.
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 “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 104753 · 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. 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…
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. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
“Having four sides” · Deductive & contrapositive…
5. 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…
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. 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…
8. What single counterexample disproves the universal claim “All prime numbers are odd”?
What single counterexample… · Deductive & contrapositive…
9. 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…
10. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
4 · 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 “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 104753 · 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. 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.
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. 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).
5. 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.
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. 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.
8. 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.
9. 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.
10. 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.
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 “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 104753 · Age guidance, not curriculum certification