Fill in the blanks in the following proof: Theorem 1. The set of prime numbers is infinite. Proof. Suppose not. Then there is a complete, finite, which con- tains all of the prime numbers. P1 = 2, P2 = 3, P3 = 5, P4 = 7, ..., Pn Consider the integer formed by taking the product of all of these primes and then adding 1. N = (piP2P3P4. .- Pn) +1 This number is . than any of the primes in our list, in particular it cannot be prime itself and must therefore be divisible by at least one of the primes (say p;). On the other hand, N is one greater than a multiple of p;, so p; cannot divide N. This is a since we have shown that N is both divisible by p; and not divisible by pj. Hence the supposition is and the theorem is true.

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.