0
0

Contrapositive Deduction

Introduction

In logical reasoning, every “If-Then” statement has an equivalent form called its contrapositive. Understanding contrapositive deduction is crucial because it helps identify equivalent statements that are always true when the original is true.

This pattern strengthens reasoning by teaching how to reverse and negate conditionals accurately - a common feature in competitive exams and analytical logic problems.

Pattern: Contrapositive Deduction

Pattern

The contrapositive of “If P → Q” is “If not Q → not P.” Both statements are logically equivalent.

That means if the original conditional is true, its contrapositive must also be true. However, its converse (“If Q → P”) and inverse (“If not P → not Q”) are not logically equivalent.

Step-by-Step Example

Question

Statement:
If it rains, the streets get wet.

Which of the following represents the contrapositive of this statement?

Options:
A. If the streets get wet, it rains.
B. If the streets do not get wet, it did not rain.
C. If it does not rain, the streets do not get wet.
D. If the streets get dry, it rains.

Solution

  1. Step 1: Identify the original conditional

    “If it rains (P) → Streets get wet (Q).”

  2. Step 2: Apply the contrapositive rule

    Contrapositive is formed by reversing and negating both parts: “If not Q → not P.”

  3. Step 3: Substitute terms

    “If the streets do not get wet (¬Q), then it did not rain (¬P).”

  4. Step 4: Match with options

    Option B represents this exactly.

  5. Final Answer:

    If the streets do not get wet, it did not rain → Option B

  6. Quick Check:

    Original and contrapositive are logically equivalent - both true or both false ✅

Quick Variations

1. “If A → B” ⇒ Contrapositive: “If not B → not A.”

2. “If you study → you pass.” ⇒ Contrapositive: “If you don’t pass → you didn’t study.”

3. “If the alarm rings → you wake up.” ⇒ Contrapositive: “If you didn’t wake up → the alarm didn’t ring.”

4. Contrapositive always maintains logical equivalence, unlike converse or inverse.

Trick to Always Use

  • Step 1: Reverse the order (make consequence first).
  • Step 2: Negate both parts (add “not” to both).
  • Step 3: Keep “If-Then” structure intact to form valid contrapositive.

Summary

Summary

  • The contrapositive of “If P → Q” is “If not Q → not P.”
  • Contrapositive and the original conditional are logically equivalent.
  • Converse (“If Q → P”) and inverse (“If not P → not Q”) are not logically valid.
  • Always check reversal and negation correctly - this avoids confusion in reasoning questions.

Example to remember:
Statement: If a person is honest, he is trusted.
Contrapositive: If a person is not trusted, he is not honest. ✅

Practice

(1/5)
1. Statement: If a student studies hard, he will pass the exam. Which of the following represents the contrapositive of this statement?
easy
A. If a student does not study hard, he will not pass the exam.
B. If a student passes the exam, he studied hard.
C. If a student does not pass the exam, he did not study hard.
D. If a student passes, he studied little.

Solution

  1. Step 1: Identify original

    ‘If studies hard (P) → passes exam (Q).’

  2. Step 2: Apply contrapositive rule

    Reverse and negate: If not Q → not P.

  3. Step 3: Substitute terms

    If student does not pass (¬Q), then he did not study hard (¬P).

  4. Final Answer:

    If a student does not pass the exam, he did not study hard → Option C

  5. Quick Check:

    Original and contrapositive both always hold true together ✅
Hint: Contrapositive = Reverse + Negate both statements.
Common Mistakes: Confusing contrapositive with inverse (‘If not P → not Q’).
2. Statement: If it is a car, it has wheels. Which of the following is the correct contrapositive?
easy
A. If it has wheels, it is a car.
B. If it is not a car, it does not have wheels.
C. If it does not have wheels, it is not a car.
D. If it is a vehicle, it has wheels.

Solution

  1. Step 1: Original form

    ‘If car (P) → has wheels (Q).’

  2. Step 2: Reverse and negate

    Contrapositive: If not Q → not P.

  3. Step 3: Substitute terms

    If it does not have wheels (¬Q), then it is not a car (¬P).

  4. Final Answer:

    If it does not have wheels, it is not a car → Option C

  5. Quick Check:

    Reversed and negated correctly ✅
Hint: When forming contrapositive, swap and add negation to both sides.
Common Mistakes: Choosing the converse (‘If it has wheels → it is a car’).
3. Statement: If a number is even, it is divisible by 2. Identify its contrapositive.
easy
A. If a number is not divisible by 2, it is not even.
B. If a number is not even, it is not divisible by 2.
C. If a number is divisible by 2, it is even.
D. If a number is odd, it is divisible by 2.

Solution

  1. Step 1: Identify original

    ‘If even (P) → divisible by 2 (Q).’

  2. Step 2: Reverse and negate

    Contrapositive: If not Q → not P.

  3. Step 3: Substitute terms

    If number is not divisible by 2 (¬Q), it is not even (¬P).

  4. Final Answer:

    If a number is not divisible by 2, it is not even → Option A

  5. Quick Check:

    Perfect logical reversal and negation ✅
Hint: ‘Not divisible by 2 → Not even’ is the contrapositive of ‘Even → Divisible by 2’.
Common Mistakes: Mistaking contrapositive for inverse or converse.
4. Statement: If a person is honest, he will be trusted. Which of the following is its contrapositive?
medium
A. If a person is trusted, he is honest.
B. If a person is not honest, he is not trusted.
C. If a person is not trusted, he is not honest.
D. If a person is dishonest, he will be trusted.

Solution

  1. Step 1: Original form

    ‘If honest (P) → trusted (Q).’

  2. Step 2: Contrapositive rule

    If not Q → not P.

  3. Step 3: Substitute

    If person is not trusted (¬Q), then he is not honest (¬P).

  4. Final Answer:

    If a person is not trusted, he is not honest → Option C

  5. Quick Check:

    Equivalent to the original conditional ✅
Hint: Always reverse and negate - not Q → not P.
Common Mistakes: Selecting inverse (‘If not P → not Q’) instead of contrapositive.
5. Statement: If the train arrives late, passengers will miss the bus. What is the contrapositive?
medium
A. If passengers miss the bus, the train arrived late.
B. If passengers did not miss the bus, the train did not arrive late.
C. If passengers miss the bus, they arrived early.
D. If train did not arrive late, passengers missed the bus.

Solution

  1. Step 1: Identify statement

    ‘If train late (P) → passengers miss bus (Q).’

  2. Step 2: Reverse and negate

    Contrapositive = If not Q → not P.

  3. Step 3: Substitute

    If passengers did not miss the bus (¬Q), then train did not arrive late (¬P).

  4. Final Answer:

    If passengers did not miss the bus, the train did not arrive late → Option B

  5. Quick Check:

    Reversed + negated properly ✅
Hint: For contrapositive, reverse order and negate both statements.
Common Mistakes: Choosing the converse (‘If Q → P’) instead of contrapositive (‘If not Q → not P’).

Mock Test

Ready for a challenge?

Take a 10-minute AI-powered test with 10 questions (Easy-Medium-Hard mix) and get instant SWOT analysis of your performance!

10 Questions
5 Minutes