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 → 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…
2. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
4 · Deductive & contrapositive…
3. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
“1” · “3” · 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. 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…
6. 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…
7. 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…
8. 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…
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. 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…
11. 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. 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 104745 · 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 → 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.
2. 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.
3. 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).
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. 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.
6. 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.
7. 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”).
8. 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.
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. 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.
11. 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. 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 104745 · Age guidance, not curriculum certification

Worksheet StudioPractice pack · Ages 16-18

Deductive & contrapositive logic

Take your time. Show your thinking.

1. 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…
2. Is “Being a multiple of 4” a sufficient condition or only a necessary condition for “Being an even number”?
4 · Deductive & contrapositive…
3. To disprove “For every integer n, n² > n”, which value of n serves as a valid counterexample: “1” or “3”?
“1” · “3” · 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. 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…
6. 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…
7. 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…
8. 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…
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. 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…
11. 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. 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 104745 · Age guidance, not curriculum certification

Worksheet StudioAnswer key · Ages 16-18

Deductive & contrapositive logic

Answers and reasoning. Review before teaching.

1. 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.
2. 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.
3. 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).
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. 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.
6. 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.
7. 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”).
8. 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.
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. 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.
11. 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. 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 104745 · Age guidance, not curriculum certification