Problem Solving and Reasoning

• Inductive Reasoning

  • A method where you reach a general conclusion based on specific examples.
• Inductive Reasoning

  • It involves observing patterns and making conjectures (predictions).
• Deductive Reasoning

  • A method where you apply general principles to reach a specific conclusion.
• Deductive Reasoning

  • It is based on rules, facts, or known properties.
• Intuition

  • A gut feeling about something being correct without formal proof.
• Indian mathematician Srinivasa Ramanujan discovered complex formulas without proof, but they were later verified.
• Proof

  • A way to logically verify that a statement is always true.
• Three main types of proofs:

  1. Direct Proof
  2. Proof by Contradiction
  3. Proof by Induction
• Direct Proof – Uses if-then logic.

  • Example: If a fruit is ripe, then it tastes sweet.
• Proof by Contradiction – Assumes a statement is false, then proves it leads to a contradiction.
• Proof by Induction – Proves a general rule by testing it for 𝑛 = 1, then assuming it holds for 𝑛 = k and proving it for 𝑛 = k+1.
• Certainty

  • Being absolutely sure about a mathematical statement.
  • Mathematicians constantly refine theorems to ensure correctness.
• Polya’s Four Steps in Problem Solving

  Mathematician George Polya developed a structured way to solve problems.

  Step 1: Understand the Problem

  Step 2: Devise a Plan

  Step 3: Carry Out the Plan

  Step 4: Look Back
• Inductive reasoning involves making general conclusions based on specific observations.
• This statement is an example of Deductive Reasoning - Noticing that all observed apples in a basket are red, so concluding that all apples are red.
• The first step in Polya’s method is to fully grasp the problem before solving it.
• Proof by Contradiction

  • This method assumes a statement is false and finds a contradiction.
• Intuition is just a feeling—it doesn’t guarantee correctness without proof.
• Deductive reasoning starts with a general rule and applies it to specific cases (not the other way around).
• Proof by induction is often used in mathematical proofs, especially for sequences and series.
• The final step in Polya’s method is "Look Back", where you check the solution and verify its correctness.
• A conjecture is a statement that has not yet been proven. Once proven, it becomes a theorem.