3. PROOF BY COUNTER EXAMPLE
• ONE OF THE VARIOUS METHODS.
• USES EXAMPLE TO DEFY THE GIVEN STATEMENT (CONJECTURE)
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CONT
4. EXAMPLE
• TO PROVE V X P(X) -> IS FALSE
- SOME VALUES OF X THAT IS FALSE
EX: PROVE OR DISPROVE PRODUCT OF TWO IRRATIONAL NUMBERS
IS IRRATIONAL.
√2 * √2 = √4 = 2 RATIONAL NUMBER
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5. MATHEMATICAL INDUCTION
• MOST IMPORTANT REASONING THAT CONSIDERS POSITIVE
INTEGERS.
• THROUGH EXPERIMENTS & OBSERVATIONS.
• TYPES
• WEAK INDUCTION
• STRONG INDUCTION
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6. WEAK INDUCTION
• FIRST PRINCIPLE OF MATHEMATICAL INDUCTION
• STEPS
1. BASIC STEP : { SHOW P(0) / P(1) IS TRUE }
2. INDUCTIVE HYPOTHESIS :
ASSUME P(K) IS TRUE.
K IS ANY ARBITRARY NUMBER.
3. INDUCTIVE STEP :
SHOW THAT P(K+1) IS TRUE ON THE
BASIS OF INDUCTIVE HYPOTHESIS.
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8. STRONG INDUCTION
• 2ND PRINCIPLE OF MATHEMATICAL INDUCTION.
• USES DIFFERENT INDUCTIVE STEP THAN FIRST PRINCIPLE.
• STEPS
1. BASIC STEP : { SHOW P(0) IS TRUE }
2. INDUCTIVE HYPOTHESIS :
ASSUME P(K) IS TRUE FOR ALL NO ≤ K ≤ N.
3. INDUCTIVE STEP :
SHOW BASED ON ASSUMPTION THAT P(K+1) IS
TRUE.
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