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Contrapositive Deduction

Introduction

Logical reasoning में हर “If-Then” statement का एक contrapositive रूप होता है, जो मूल statement जितना ही valid और हमेशा logically equivalent होता है।

Contrapositive deduction आपको conditionals को सही तरह से reverse और negate करना सिखाता है - जो competitive exams और analytical logic problems में एक बहुत ही महत्वपूर्ण skill है।

Pattern: Contrapositive Deduction

Pattern

“If P → Q” का contrapositive है “If not Q → not P” - दोनों हमेशा logically equivalent होते हैं।

इसका मतलब है कि अगर original statement true है, तो उसका contrapositive भी हर स्थिति में true होगा। लेकिन इसका converse (“If Q → P”) और inverse (“If not P → not Q”) logically equivalent नहीं होते - ये common traps हैं।

Step-by-Step Example

Question

Statement:
If it rains, the streets get wet.

Which of the following represents the contrapositive?

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: Original conditional पहचानें

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

  2. Step 2: Contrapositive rule लागू करें

    Reverse + Negate: “If not Q → not P.”

  3. Step 3: Terms substitute करें

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

  4. Step 4: Options चेक करें

    यह Option B से perfectly match होता है।

  5. Final Answer:

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

  6. Quick Check:

    Original और contrapositive हमेशा एकसाथ true या false होते हैं।

Quick Variations

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

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

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

4. याद रखें: Contrapositive ही एकमात्र logically equivalent form है - converse और inverse नहीं।

Trick to Always Use

  • Step 1: Statement को reverse करें (Q पहले, P बाद में)।
  • Step 2: दोनों parts को negate करें (“not” जोड़ें)।
  • Step 3: “If-Then” structure intact रखें - तभी valid contrapositive बनेगा।

Summary

Summary

  • “If P → Q” का contrapositive हमेशा “If not Q → not P” होता है।
  • Original और contrapositive दोनों logically equivalent होते हैं।
  • Converse और inverse traps हैं - ये logically equivalent नहीं होते।
  • Contrapositive बनाते समय reversal + negation सही तरह से करना आवश्यक है।

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

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