Please provide a step-by-step answer with a full explanation about every step in

ID: 2986181 • Letter: P

Question

Please provide a step-by-step answer with a full explanation about every step involved for maximum points. Thank you.

Give an example of a predicate P(n) about positive integers n, such that P(n) is

true for every positive integer from 1 to one billion, but which is never-the-less not

true for all positive integers.

(Hints: (1) There is a really simple choice possible

for the predicate P(n), (2) Make sure you write down a predicate with variable n!)

The purpose of this problem is to convince you that when checking a for all type

proposition, it is not good enough to just check the truth for a few sample cases,

or, for that matter, even a few billion sample cases.

A general proof that covers all possible cases is necessary.

Predicate : P(n) is greater than zero

and

P(n) is defined as

P(n) = 1+n ........ if (1<n < one billion )

P(n) = 1-n ....... if (n> one billion )

so,

P(n) will be true for all n between 1 and one billion but not for the all the positive integers

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