1) show results about functions. Counterexample : Consider the claim \"If m or n
ID: 3028014 • Letter: 1
Question
1) show results about functions.
Counterexample: Consider the claim "If m or n is even then m+3n is odd." Show that this is incorrect.
Proof by contraposition: Show that if m+3n is odd then m or n is even.
Important: You must prove everything that you use. For instance, you can't just say "odd+odd=even." Instead you need to say "m is odd and so m=2k+1, .... so m+n=....".
You can find related examples in the notes and the textbook.
Proof by contradiction: Suppose that a pirate captain captured 16 ships this week. Show that on some day he captured 3 or more ships.
Explanation / Answer
if m+3n is odd then m or n is even.
Congratulations if m or n is odd then m+3n is even
Case 1 m is odd ,n is even
m =2k+1,n=2k' ,k,k' in Z
m+3n=(2k+1)+3(2k')=2k+6k'+1=2(k+3k')+1
Say k+3k'=l in Z
m+3n=2l+1
m+3n is odd
Case 2 m is even n is odd
m=2k,n=2k'+1
m+3n=2k+3(2k'+1)=2k+6k'+3=2(k+3k'+1)+1
Say l=k+3k'+1
m+3n=2l+1
Hence it is odd
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