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 December 1st, 2011, 10:25 AM #1 Newbie   Joined: Dec 2011 Posts: 15 Thanks: 0 Direct Proof? Let a, b belong to Z. Prove that ab is even if and only if a is even or b is even. December 1st, 2011, 01:17 PM   #2
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Re: Direct Proof?

Quote:
 Originally Posted by jrklx250s Let a, b belong to Z. Prove that ab is even if and only if a is even or b is even.
By contrapositive direct proof : if a ,is not even and b is not even then ab is not even.

I think this easy to prove.

Then you can easily prove ; if a ,is even and b, is even then ab is even December 2nd, 2011, 04:04 PM #3 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Direct Proof? Well, I doubt that a direct proof of the contrapositive would be considered a direct proof of the theorem itself. A number is even if and only if it has a two as a prime factor. If a does not have a factor of 2, then b must. Therefore either a is even or b is even. Notice that the fact that 2 is a prime is important. It is, for example, NOT true that "if ab is dividisible by 6 then either a is divisible by 6 or b is divisible by 6". December 3rd, 2011, 03:58 AM   #4
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Re: Direct Proof?

Quote:
 Originally Posted by HallsofIvy Well, I doubt that a direct proof of the contrapositive would be considered a direct proof of the theorem itself. A number is even if and only if it has a two as a prime factor. If a does not have a factor of 2, then b must. Therefore either a is even or b is even. Notice that the fact that 2 is a prime is important. It is, for example, NOT true that "if ab is dividisible by 6 then either a is divisible by 6 or b is divisible by 6".
An indirect proof is only a proof by contradiction

By logic we have : p=>q is equivalent to :~q =>~p ,which is equivalent to: ~p v q ([color=#FF0000]which you are using in your proof)[/color].

Don't you contradict your self ?? Tags direct, proof Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post shaharhada Abstract Algebra 7 January 23rd, 2014 06:34 PM mgk501 Real Analysis 2 March 21st, 2013 12:23 PM MathematicallyObtuse Algebra 4 January 15th, 2011 07:26 PM gianni Abstract Algebra 2 December 2nd, 2010 05:00 AM MathNoob Calculus 4 March 5th, 2009 06:18 AM

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