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4 10 Marks Working Infinity Sometimes Dealing Predicates N Don T Care Much Numbers Satisfy Q43883928

expression about prime

4. (10 marks] Working with infinity. Sometimes when dealing with predicates over N, we don’t care so much about which numbers satisfy the given predicate, or even exactly how many numbers satisfy it. Instead, we care about whether the predicate is satisfied by infinitely many numbers. For example: “There are infinitely many primes.” We saw that for the natural numbers, we can express the idea of “infinitely many” by saying that for every natural number no, there is a number greater than no that satisfies the predicate: Vno E N En EN n > no A Prime(n) You may use the Prime predicate in your solutions for all parts of this question. (a) Express the following statement in predicate logic: “There are finitely many natural numbers that are not prime.” (b) Consider the following definition. Definition 2. Let a E N. We say that a is a prime gap when there exists a prime p such that p+a is also prime, and none of the numbers between p and p+a (exclusive) are prime. Translate the following statement into predicate logic: “There are infinitely many prime gaps.” You may not define your own predicate for “Prime Gap”; here, we’re looking for you to translate the above definition and embed that translation into a larger logical statement. (c) Here is another variation of combining predicates with infinity. Let P:N → True, False}. We say that P is eventually true when there exists a natural number no such that all natural numbers greater than no satisfy P. Use this idea to express the following statement in predicate logic (no justification is required here): Eventually, all natural numbers are not prime gaps. Show transcribed image text 4. (10 marks] Working with infinity. Sometimes when dealing with predicates over N, we don’t care so much about which numbers satisfy the given predicate, or even exactly how many numbers satisfy it. Instead, we care about whether the predicate is satisfied by infinitely many numbers. For example: “There are infinitely many primes.” We saw that for the natural numbers, we can express the idea of “infinitely many” by saying that for every natural number no, there is a number greater than no that satisfies the predicate: Vno E N En EN n > no A Prime(n) You may use the Prime predicate in your solutions for all parts of this question. (a) Express the following statement in predicate logic: “There are finitely many natural numbers that are not prime.” (b) Consider the following definition. Definition 2. Let a E N. We say that a is a prime gap when there exists a prime p such that p+a is also prime, and none of the numbers between p and p+a (exclusive) are prime. Translate the following statement into predicate logic: “There are infinitely many prime gaps.” You may not define your own predicate for “Prime Gap”; here, we’re looking for you to translate the above definition and embed that translation into a larger logical statement. (c) Here is another variation of combining predicates with infinity. Let P:N → True, False}. We say that P is eventually true when there exists a natural number no such that all natural numbers greater than no satisfy P. Use this idea to express the following statement in predicate logic (no justification is required here): Eventually, all natural numbers are not prime gaps.

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Answer to 4. (10 marks] Working with infinity. Sometimes when dealing with predicates over N, we don’t care so much about which nu…