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  
Senior Member Joined: Apr 2010 Posts: 451 Thanks: 1  Re: Direct Proof? Quote:
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  
Senior Member Joined: Apr 2010 Posts: 451 Thanks: 1  Re: Direct Proof? Quote:
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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
direct product  shaharhada  Abstract Algebra  7  January 23rd, 2014 06:34 PM 
Discrete math direct proof question  mgk501  Real Analysis  2  March 21st, 2013 12:23 PM 
Direct proportion  MathematicallyObtuse  Algebra  4  January 15th, 2011 07:26 PM 
Direct product  gianni  Abstract Algebra  2  December 2nd, 2010 05:00 AM 
Direct Substitution?  MathNoob  Calculus  4  March 5th, 2009 06:18 AM 