When do you use regular induction vs strong induction?
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.
When do you use the strong principle of mathematical 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.
Is strong induction equivalent to induction?
We can conclude, via strong induction, that the statement holds for all positive integers n, but this is the exact same conclusion that regular induction would have. Thus, regular induction will hold whenever strong induction holds.
Does strong induction equal weak induction?
Proof:Strong induction is equivalent to weak induction.
Is strong induction more powerful?
Despite the name, strong induction is actually no more powerful than ordinary induction. In other words, any theorem that can be proved with strong induction could also be proved with ordinary induction (using a slightly more complicated indcution hypothesis). But strong induction can make some proofs a bit easier.
What is simple induction?
(Simple induction is sometimes also called weak induction.) In a simple induction proof, we prove two parts. Part 2 — Induction Step: ∀i ≥ 0, (P(i) → P(i + 1)). Note 1: Informally, part 2 says, P(0) implies P(1), P(1) implies P(2), P(2) implies P(3), and so on. Formally, there are other ways to say the same thing.
Why is strong induction necessary?
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.
Is strong induction equivalent weak induction?
What is the difference between 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.
When to use 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. How do you get a strong induction? To prove this using strong induction, we do the following:
What to expect from an induction?
In a pregnancy that is progressing normally, your body and your baby’s secrete the hormone oxytocin, triggering labor. This starts contractions and preps your cervix by thinning and softening it. Induction is an attempt to jump-start this process.
What are the fallacies of weak induction?
Fallacies of weak induction are fallacies where there is a weak connection between the premises and the conclusion. Inductive arguments are arguments where the conclusions are improbably true if the premises are true.
What is a weak induction?
Weak Induction. Weak induction is used to show that a given property holds for all members of a countable inductive set, this usually is used for the set of natural numbers. Weak induction for proving a statement P ( n ) {\\displaystyle P(n)} (that depends on n {\\displaystyle n} ) relies on two steps: