Problem Solving and Reasoning

Cards (22)

  • 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.