What is strong induction in discrete mathematics?

Strong induction is a variant of induction, in which we assume that the statement holds for all values preceding k. This provides us with more information to use when trying to prove the statement.

What is strong and weak induction?

The difference between weak induction and strong indcution only appears in induction hypothesis. In weak induction, we only assume that particular statement holds at k-th step, while in strong induction, we assume that the particular statment holds at all the steps from the base case to k-th step.

What is the principle of strong induction?

Principle of Strong Mathematical Induction: If P is a set of integers such that (i) a is in P, (ii) if all integers k, with a ≤ k ≤ n are in P, then the integer n + 1 is also in P, then P = {x ∈ Z | x ≥ a} that is, P is the set of all integers greater than or equal to a. Theorem.

What is the point of strong induction?

This variant of an induction proof is called “strong induction.” A standard application of strong induction (with the induction hypothesis being “P(k −1) and P(k)” instead of just “P(k)”) is to proving identities and relations for Fibonacci numbers and other recurrences.

When can you use strong induction?

2 Answers. Show activity on this post. With simple induction you use “if p(k) is true then p(k+1) is true” while in strong induction you use “if p(i) is true for all i less than or equal to k then p(k+1) is true”, where p(k) is some statement depending on the positive integer k.

What is a weak induction?

Whenever someone concludes something from premises that only provide a very WEAK support for that conclusion, they are being irrational, and committing an informal fallacy; namely, a “fallacy of weak induction”.

Is strong and weak induction equivalent?

You may think that strong induction is stronger than weak induction in the sense that you can prove more things using strong induction than you could using only weak induction (the names certainly suggest that!).

What is the principle of induction in math?

The principle of mathematical induction is then: If the integer 0 belongs to the class F and F is hereditary, every nonnegative integer belongs to F. Alternatively, if the integer 1 belongs to the class F and F is hereditary, then every positive integer belongs to F.

Is strong induction equivalent to weak induction?

What is inductive hypothesis in discrete mathematics?

[Inductive Hypothesis] If the truth of P(k) implies the truth of P(k + 1), then the statement P(n) is true for all n ≥ a.