Final answer:
To prove the implication that prime numbers are solitary, one can embark on a direct proof, a proof by contrapositive, or a proof by contradiction, by starting with the relevant assumption and logically proceeding to show that the statement must hold true.
Step-by-step explanation:
To prove the implication "For all numbers n, if n is prime then n is solitary," we can approach it in different ways, depending on the proof strategy we decide to use. Here are the initial and concluding parts of such proofs, not including the definitions of the terms or the detailed steps, which are based on the logic of valid deductive inferences.
Begin by assuming that we have a number n which is prime. Since being prime is the sufficient condition for our assertion, explicitly state that such an n must then show the solitary property...
After going through logical steps and showing that prime numbers indeed exhibit the property of being solitary, we therefore conclude that for all numbers n, if n is prime, then n is solitary.
To prove this by contrapositive, we start by assuming that n is not solitary. This assumption serves as our beginning premise...
If these steps lead us to the conclusion that n cannot be prime, we would have successfully shown the contrapositive: if n is not solitary, then n is not prime, and therefore the original statement must be true.
For a proof by contradiction, let us assume that the statement is false, that is, there exists a prime number n that is not solitary...
If this leads us to a contradiction with any known properties of prime numbers or solitary nature, then our assumption that the statement is false is incorrect, and thus the original implication stands as true.