Expert: Paul Klarreich - 11/8/2008

Question
Prove by contrapositive. Let x be an element of the set of integers. If 5x-7 is even, then x is odd.

I believe that you start by assuming that x is even, therefore x = 2k and assume that 5x-7 is odd then substitute 2k into the equation 5x-7. Does that sound right so far? What is next?

Answer
Questioner:   Pete
Category:  Advanced Math
Private:  No
 
Subject:  Proofs
Question:  Prove by contrapositive. Let x be an element of the set of integers. If 5x-7 is even, then x is odd.

I believe that you start by assuming that x is even, therefore x = 2k and assume that 5x-7 is odd then substitute 2k into the equation 5x-7. Does that sound right so far? What is next?
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Hi, Pete,

I believe you say 'proof by contradiction', but let it pass.  

To prove: If THIS, then THAT.

First: assume THAT is false.
Then prove: THIS  is false.

I believe that you start by assuming that x is even,

>> exactly so.

therefore x = 2k

>> therefore  x CAN BE WRITTEN 2k.

and assume that 5x-7 is odd then substitute 2k into the equation 5x-7.

>> 5x - 7 is not an equation.  Use vocabulary carefully.

Does that sound right so far? What is next?

>> Well, do it!

You want to prove 5x - 7 = 5(2k) - 7 is odd.

Now to show something is odd, show that it CAN BE WRITTEN:

2(some-stuff-that-is-an-integer) + 1.

So do some algebra to massage  5(2k) - 7 into that form.

Like:  5(2k) - 7 = 10k - 7 = 10k - 6 - 1 =

2(5x - 3) - 1 = OOPS, we wanted 2(something) PLUS 1.

Can you see how to fix it and finish?