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. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
“1” · “3” · Deductive & contrapositive…
2. 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…
3. 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…
4. 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…
5. 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…
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 “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…
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 “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…
10. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
“Having four sides” · Deductive & contrapositive…
11. 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…
12. What single counterexample disproves the universal claim “All prime numbers are odd”?
What single counterexample… · Deductive & contrapositive…

12 questions · Seed 104732 · 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. 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).
2. 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.
3. 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.
4. 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.
5. 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.
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 “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.
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 “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.
10. 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).
11. 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.
12. 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.

12 questions · Seed 104732 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Deductive & contrapositive logic

Take your time. Show your thinking.

1. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
“1” · “3” · Deductive & contrapositive…
2. 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…
3. 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…
4. 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…
5. 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…
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 “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…
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 “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…
10. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
“Having four sides” · Deductive & contrapositive…
11. 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…
12. What single counterexample disproves the universal claim “All prime numbers are odd”?
What single counterexample… · Deductive & contrapositive…

12 questions · Seed 104732 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Deductive & contrapositive logic

Answers and reasoning. Review before teaching.

1. 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).
2. 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.
3. 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.
4. 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.
5. 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.
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 “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.
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 “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.
10. 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).
11. 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.
12. 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.

12 questions · Seed 104732 · Age guidance, not curriculum certification