URGENT !!! CORRECT AND FULL ANSWERS ONLY Instructions: Prove the following theor

Question

URGENT !!! CORRECT AND FULL ANSWERS ONLY

Instructions: Prove the following theorems. Write your proofs in complete sentences and make them self contained. While existential statements may be proven by providing a single example of an object satisfying the predicate, universal statements (on infinite sets) can’t be proven by examples. Use direct proof methods to prove such universal statements. Don’t assume results shown in class; i.e. prove the universal statements from scratch using only definitions and proper proof technique.

Theorem 1: There exists an integer m Z such that (3m + 4)/m = 2.

Theorem 2: Suppose a and b are integers. If a is odd and b is even then a+b is odd.

Theorem 3: Suppose a is an integer. If a is odd, then 2 a^2 is odd.

Explanation / Answer

Theorem 1: There exists an integer m Z such that (3m + 4)/m = 2

Proof: Given that (3m + 4)/m = 2

=> 3m + 4 = 2m

=> 3m - 2m = 4

=> m = 4

Since m Z, the given theorem is proved.

2. Suppose a and b are integers. If a is odd and b is even then a+b is odd.

Proof: Since a is odd, let a = 2k + 1 where k is an integer.

Since b is even, let b = 2l where l is an integer.

=> a + b = 2k + 1 + 2l

=> a + b = 2k + 2l + 1

=> a + b = 2 (k + l) + 1

Let k + l = m where m is an integer.

=> a + b = 2l + 1

Since a + b is of the form 2l + 1, it is odd.

Theorem 3: Suppose a is an integer. If a is odd, then 2 a2 is odd.

Proof: Since a is odd, let a = 2k + 1 where k is an integer

=> 2 - a2 = 2 - (2k + 1)2

=> 2 - a2 = 2 - (4k2 + 4k + 1)

=> 2 - a2 = 2 - 4k2 - 4k - 1

=> 2 - a2 = - 4k2 - 4k + 1

=> 2 - a2 = 2 (- 2k2 - 2k) + 1

Let - 2k2 - 2k = r where r is an integer

=> 2 - a2 = 2r + 1

Since 2 - a2 is of the form 2r + 1, it is odd.

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