Strong Induction
Assume all earlier cases to prove the next.
Key idea
Assume all earlier cases to prove the next.
In depth
Assume P holds for all values up to k, then prove P(k + 1). Useful when the next case depends on several earlier ones, such as proving every integer above 1 is a product of primes.
Worked example
A number is prime, or splits as ab with a, b < n, each a product of primes by the hypothesis.
Real-world uses
- Proving properties of recursive structures such as trees and stamp combinations
Common mistake
Forgetting to handle enough base cases.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Strong Induction?
Assume all earlier cases to prove the next. It is a topic at the expert level in Discrete Math & Number Theory.
Where is Strong Induction used in real life?
Real-world examples: Proving properties of recursive structures such as trees and stamp combinations.
What should I learn before Strong Induction?
Make sure you are comfortable with Induction Principle, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Strong Induction?
Forgetting to handle enough base cases.
How can I practise Strong Induction?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Strong Induction?
Move on to Well-Ordering Principle next.
Related expert topics in Discrete Math & Number Theory
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Cardinality of Sets – Comparing sizes of infinite sets.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Cantor’s Diagonal Argument – Proof that real numbers are uncountable.
- Combinations with Repetition – Stars and bars.