Save
...
1st Semester - BSIT
Finals - MMW
Problem Solving and Reasoning
Save
Share
Learn
Content
Leaderboard
Share
Learn
Created by
Marc
Visit profile
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:
Direct Proof
Proof by Contradiction
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.