Student A is faced with having to prove the claim forall n greaterthanequalto^N
ID: 3108848 • Letter: S
Question
Student A is faced with having to prove the claim forall n greaterthanequalto^N 5: P(n), where P(x) is a given predicate with domain {5, 6, 7, ellipsis}. Student B declares that to prove (1) it is sufficient to prove the following two claims: 1. P(5) is True; 2. forall n greaterthanequalto^N 5: ((P(5) And P(6) And P(7) And ellipsis And P(n)) p(n + 1)). 1. Describe in common language what it is that the two claims are stating, in terms of "lists of statements". 2. Imitate the proof presented in class to argue that Student B is indeed correct. 3. Explain, why one has nothing much to lose and has much to gain in always going with the method of complete induction, rather than the method of induction.Explanation / Answer
The principle of induction
Let P(n) be a predicate,if
(a) P(0) is true
(b) P(n) implies P(n+1) for all nonnegative integers n.
then, (c) P(m) is true for all non negative integers m.