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Contrapositive Deduction

Introduction

Logical reasoning இல், ஒவ்வொரு “If-Then” கூற்றுக்கும் அதற்கு இணையான ஒரு வடிவம் உள்ளது; அதையே contrapositive என்கிறோம். Contrapositive deduction என்பதை புரிந்துகொள்வது மிகவும் முக்கியம், ஏனெனில் அசல் கூற்று உண்மையாக இருந்தால் எப்போதும் உண்மையாக இருக்கும் சமமான கூற்றுகளை இது கண்டறிய உதவுகிறது.

இந்த pattern, conditionals-ஐ சரியாக மாற்றி (reverse) மறுப்பு (negate) செய்வதை கற்றுத்தருவதன் மூலம் reasoning திறனை வலுப்படுத்துகிறது - இது competitive exams மற்றும் analytical logic questions இல் அடிக்கடி காணப்படும் அம்சமாகும்.

Pattern: Contrapositive Deduction

Pattern

“If P → Q” என்பதின் contrapositive என்பது “If not Q → not P.” ஆகும். இந்த இரண்டும் logical-ஆக சமமானவை.

அதாவது, அசல் conditional உண்மையாக இருந்தால், அதன் contrapositive கூட கட்டாயமாக உண்மையாக இருக்கும். ஆனால், அதன் converse (“If Q → P”) மற்றும் inverse (“If not P → not Q”) ஆகியவை logical-ஆக சமமானவை அல்ல.

Step-by-Step Example

Question

Statement:
If it rains, the streets get wet.

கீழ்கண்டவற்றில் எது இந்த statement-இன் contrapositive-ஐ சரியாக காட்டுகிறது?

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 என்பது இரு பகுதிகளையும் மாற்றி (reverse) மறுப்பு செய்வதன் மூலம் உருவாகும்: “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 இதைத் துல்லியமாகக் காட்டுகிறது.

  5. Final Answer:

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

  6. Quick Check:

    Original மற்றும் contrapositive இரண்டும் logical-ஆக சமமானவை - இரண்டும் ஒரே நேரத்தில் true அல்லது false ஆகும் ✅

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 எப்போதும் logical equivalence-ஐ பாதுகாக்கும்; converse அல்லது inverse அப்படியல்ல.

Trick to Always Use

  • Step 1: வரிசையை மாற்றுங்கள் (consequence-ஐ முதலில் கொண்டு வாருங்கள்).
  • Step 2: இரு பகுதிகளையும் negate செய்யுங்கள் (“not” சேர்க்கவும்).
  • Step 3: சரியான contrapositive உருவாக “If-Then” அமைப்பை 그대로 வைத்திருங்கள்.

Summary

Summary

  • “If P → Q” என்பதின் contrapositive: “If not Q → not P.”
  • Contrapositive மற்றும் original conditional இரண்டும் logical-ஆக சமமானவை.
  • Converse (“If Q → P”) மற்றும் inverse (“If not P → not Q”) logical-ஆக செல்லுபடியாகாது.
  • Reversal மற்றும் negation-ஐ எப்போதும் சரியாகச் சரிபாருங்கள் - இது reasoning questions இல் குழப்பத்தை தவிர்க்க உதவும்.

நினைவில் கொள்ள வேண்டிய உதாரணம்:
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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