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. 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…
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 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…
4. 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…
5. 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…
6. 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…
7. 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…
8. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
“Having four sides” · Deductive & contrapositive…
9. 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…
10. 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…
11. 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…
12. 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…

12 questions · Seed 104744 · 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. 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.
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 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.
4. 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”).
5. 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.
6. 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.
7. 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.
8. 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).
9. 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.
10. 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.
11. 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.
12. 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.

12 questions · Seed 104744 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Deductive & contrapositive logic

Take your time. Show your thinking.

1. 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…
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 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…
4. 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…
5. 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…
6. 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…
7. 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…
8. Is “Having four sides” a necessary condition or a sufficient condition for a polygon to be a square?
“Having four sides” · Deductive & contrapositive…
9. 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…
10. 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…
11. 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…
12. 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…

12 questions · Seed 104744 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Deductive & contrapositive logic

Answers and reasoning. Review before teaching.

1. 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.
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 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.
4. 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”).
5. 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.
6. 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.
7. 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.
8. 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).
9. 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.
10. 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.
11. 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.
12. 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.

12 questions · Seed 104744 · Age guidance, not curriculum certification