What does strong induction mean?
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.
When can you use proof by induction?
Proofs by Induction A proof by induction is just like an ordinary proof in which every step must be justified. However it employs a neat trick which allows you to prove a statement about an arbitrary number n by first proving it is true when n is 1 and then assuming it is true for n=k and showing it is true for n=k+1.
Which of the following best explains the difference between weak and strong 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 many base cases are needed 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.
How do you get a strong induction?
To prove this using strong induction, we do the following:
- The base case. We prove that P(1) is true (or occasionally P(0) or some other P(n), depending on the problem).
- The induction step. We prove that if P(1), P(2), …, P(k) are all true, then P(k+1) must also be true.
Which of the following is a difference between induction and strong induction?
2 Answers. 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.
What is the first step in an induction proof?
The inductive step in a proof by induction is to show that for any choice of k, if P(k) is true, then P(k+1) is true. Typically, you’d prove this by assum- ing P(k) and then proving P(k+1). We recommend specifically writing out both what the as- sumption P(k) means and what you’re going to prove when you show P(k+1).
Is induction an axiom?
The principle of mathematical induction is usually stated as an axiom of the natural numbers; see Peano axioms. It is strictly stronger than the well-ordering principle in the context of the other Peano axioms.
Is proof by induction valid?
While this is the idea, the formal proof that mathematical induction is a valid proof technique tends to rely on the well-ordering principle of the natural numbers; namely, that every nonempty set of positive integers contains a least element. See, for example, here.
How do you write proof of induction?
Begin any induction proof by stating precisely, and prominently, the statement (“P(n)”) you plan to prove. A good idea is to put the statement in a display and label it, so that it is easy to spot, and easy to reference; see the sample proofs for examples.
How do you prove induction examples?
Proof by Induction: Further Examples
Prove by induction that 11n − 6 is divisible by 5 for every positive integer n. 11n − 6 is divisible by 5. Base Case: When n = 1 we have 111 − 6=5 which is divisible by 5. So P(1) is correct.
Does induction work on sets?
2 Answers. No, induction does not work since not every set is finite.
How do you use the well ordering principle?
The well–ordering principle says that the positive integers are well–ordered. An ordered set is said to be well–ordered if each and every nonempty subset has a smallest or least element. So the well–ordering principle is the following statement: Every nonempty subset S S S of the positive integers has a least element.
What is the principle of strong mathematical induction?
Conclusion: By the strong induction principle, it follows that P(n) is true for all n ≥ 2, i.e., every integer n ≥ 2 is either a prime or can be represented as a product of primes.