For the following, you must use good proof technique; follow the structure in th

Question

For the following, you must use good proof technique; follow the structure in the Course Notes/videos.

Clearly state what is assumed [the premises] and state reasons for each step.  

The reader must be able to easily follow your reasoning.  

When a particular proof technique is stated, you must use that proof technique.

1)

State what must be proved for the “forward proof” part of proving the following biconditional:

For any positive integer n, n is even if and only if 3n+2 is even.

Complete a DIRECT proof of the “forward proof” part of the biconditional stated in part a.

2)

State what must be proved for the “backward proof” part of proving the following biconditional:

For any positive integer n, n is even if and only if 3n+2 is even.

Complete a proof by CONTRADICTION of the “backward proof” part of the biconditional stated in part a.

3)

State the CONTRAPOSITIVE of the following conditional statement:

For all integers n, if 3n+1 is odd, then n is even.

Complete an INDIRECT proof [by proving the CONTRAPOSITIVE you wrote in part a.] to show the following statement is true. Remember to state the final conclusion after completing the proof of the contrapositive.

For all integers n, if 3n+1 is odd, then n is even.

4)

Rewrite the following statement in the form “ x, y, P(x,y)”. Clearly indicate the domain of discourse for x and y and state what the predicate P(x,y) represents.

     The product of a rational number and an irrational number is irrational.

Provide a counter-example that disproves the following proposition:

     The product of a rational number and an irrational number is irrational.

Explanation / Answer

For any positive integer n, if n is even then 3n + 2 is even.

1. The forward proof should prove that n is even given that 3n + 2 is even.

Proof: It is given that n is an even integer.

Let n = 2k where k is an integer

=> 3n + 2 = 3*2k + 2

=> 3n + 2 = 2*3k + 2

=> 3n + 2 = 2 (3k + 1)

Let l = 3k + 1 where l is an integer

=> n = 2l

=> n is even.

2. The backward proof should prove that n is even given that 3n + 2 is even.

Proof by contradiction: It is given that 3n + 2 is an even integer.

Let us assume n is odd. Let n = 2k + 1 where k is an integer.

=> 3n + 2 = 3*(2k + 1) + 2

=> 3n + 2 = 2*3k + 3 + 2

=> 3n + 2 = 2*3k + 4 + 1

=> 3n + 2 = 2 (3k + 2) + 1

Let l = 3k + 2 where l is an integer

=> 3n + 2 = 2l + 1

=> 3n + 2 is odd which contradicts with the given statement that 3n + 2 i even

Therefore by contradiction, n i even.

3. For all integers n, if 3n+1 is odd, then n is even

The contrapositive is if n is not even, 3n + 1 is not odd

Proof by contrapositive: Since n is not even, it is odd

Let n = 2k + 1 where k is an integer

=> 3n + 1 = 3(2k + 1) + 1

=> 3n + 1 = 3*2k + 3 + 1

=> 3n + 1 = 3*2k + 4

=> 3n + 1 = 2(3k + 2)

Let l = 3k + 2 where l is an integer

=> 3n + 1 = 2l

=> 3n + 1 is even

=> 3n +1 is not odd.

4. The product of a rational number and an irrational number is irrational.

For all rational numbers x and y, their product is irrational.

x, y, P(x,y)

where P(x,y) means product of x and y is irrational.

Counter-example: Let p = 0 and q = 2

p is rational and q is irrational

P(p,q) = p*q = 0*2 = 0

=> P(p,q) is rational and not irrational.

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