What is a strong induction proof?
Strong induction is a type of proof closely related to simple induction. As in simple induction, we have a statement P(n) about the whole number n, and we want to prove that P(n) is true for every value of n.
What is strong induction example?
With a strong induction, we can make the connection between P(n+1) and earlier facts in the sequence that are relevant. For example, if n+1=72, then P(36) and P(24) are useful facts. Proof: The proof is by strong induction over the natural numbers n > 1.
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.
How do you determine base case for strong induction?
Strong induction is often used where there is a recurrence relation, i.e. an=an−1−an−2. In this situation, since 2 different steps are needed to work with the given formula, you need to have at least 2 base cases to avoid any holes in your proof.
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.
Is strong induction equivalent to weak induction?
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!).
Is strong induction equivalent weak induction?
What is weak mathematical induction?
Written in predicate logic, the formula for weak mathematical induction is: (P(0) ∧ ∀k∈N[P(k) → P(k + 1)]) → ∀n∈NP(n) Given a statement P(n) defined over for all n ∈ N, to prove ∀n∈NP(n). . . 1. Prove P(0) is true. This is the Base Case.
What is the meaning of strong induction?
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.
Why is strong induction called strong?
The reason why this is called “strong induction” is that we use more statements in the inductive hypothesis. Let’s write what we’ve learned till now a bit more formally. Step 1.
How do simple induction and strong induction differ?
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. They are NOT “identical” but they are equivalent.
Why do we use strong induction?