3. Consider the following \"proofs\". Unfortunately, each proof has a mistake in

Question

3. Consider the following "proofs". Unfortunately, each proof has a mistake in it. Find the mistake in each proof, and explain why it is a mistake (you do not have to write a correct proof). [4] a. Theorem: The sum of two powers of 2 is also a power of 2. Proof: Suppose r and y are powers of 2. By the definition of being a power of 2, there exists some integer z such that x = 2: Similarly, by the definition of being a power of 2, there exists some integer z such that y = 2, Their sum, x + y = 2: +2°-2 . 2:2:+1. By the closure of the integers under addition, +1 must also be an integer. Thus the sum of x and y is a power of 2. [4] b. Theorem: If an integer n is prime, then (2" 1)/3 is a prime. 7, n Proof: Suppose that n is a prime, for instance n = 3, n n= 13. Then: 11 or

Explanation / Answer

a.) It does consider the addition of two same numbers i.e, 2z which on adding two times will give power of 2. In order to prove above inequality, we need to prove with different numbers. However, x = 20 = 1 and y = 22 contradicts the above inequality.

b.) For n = 2, (22 + 1 ) / 3 = 5/3 which is not a prime.

c.) Since theorem considers the product of 2 non-prime numbers, while in proof we assume a and b as greater and less than equal to 2.

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