Sometimes when dealing with predicates over the natural numbers, we dont care so

Question

Sometimes when dealing with predicates over the natural numbers, we dont care so much about which numbers satisfy the given predicate, or even exactly how many numbers satisfy it Instead, we care about whether infinitely many numbers satisfy the predicate. For example: "There are infinitely many primes." We can express the idea of "inifinitely many" by saying that for eveny natural number n, there is a natural number p, greater than n, that satisifes the predicate Prime (x):: "x is a prime number" Symbolically, we write: forall n element N, esists p element N, p > logicalnd Prime (p) Use the idea explained abiove to express the following statement in predicate logic (no proof required!) "There are infinitely many prime numbers of the form 4k + 1, " (ii) Write the negation of the previous symbolic statement. Push the negation symbol as far as possibl for full marks.

Explanation / Answer

(i)

For every natural number n, there is a natural number p , greater than n that satisfies the predicate prime.

Hence there are infinite prime numbers of the form 4k+1

K= 1 , prime = 5.

(ii)

We can write negation as:

~p<n/ prime(p)

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