Question

In: Advanced Math

Prove by contraposition and again by contradiction: For all integers a,b, and c, if a divides...

Prove by contraposition and again by contradiction:

For all integers a,b, and c, if a divides b and a does not divide c then a does not divide b + c

Elaboration with definitions / properties used would be appreciated!

Thanks in advance!!

Solutions

Expert Solution

Proof by Contraposition : Suppose divides .

Then there exist such that  

Now if divides then there exist such that .

is a multiple of and so divides .

So it is not true that divides and does not divides .

Hence by contraposition if   divides and does not divides then does not divides .

Proof by Contradiction : Given divides and does not divides .

Suppose   divides .

Then there exist such that  

As divides then there exist such that .

is a multiple of and so divides .

a contradiction to does not divides .

Hence if   divides and does not divides then does not divides .

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Please comment if needed .

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Colby Messinger answered 3 months ago
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