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Problem Let x, y ∈ R. Prove that if y

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1 LO Batory September 5, 2017

Problem Let x, y ∈ R. Prove that if y

3

+ yx

2

¬ x

3

+ xy

2

then y ¬ x.

Solution We want to prove a conditional statement. The equivalent to this statement is its contrapositive:

if y > x, then y

3

+ yx

2

> x

3

+ xy

2

y > x is our assumption. y

3

+ yx

2

> x

3

+ xy

2

is the conclusion we want to reach.

We start with the assumption and multiply both sides by (x

2

+ y

2

):

y > x / (x

2

+ y

2

) y(x

2

+ y

2

) > x(x

2

+ y

2

)

yx

2

+ y

3

> x

3

+ xy

2

y

3

+ yx

2

> x

3

+ xy

2

Of course the crucial step is multiplying both sides by (x

2

+ y

2

). Can we do

this? We need to make sure of two things. First, that this expression is not

less than 0 (otherwise we would need to reverse the inequality sign). Second,

that it is not 0 (if it is, then the inequality may not be strict, i.e. it would

become ­ instead of >). Think why in our example x

2

+ y

2

is positive.

Cytaty

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