How can the proof by contrapositive be used to show that x^2<x implies x<1?

[kathrynag]

Homework Statement

x^2<x, then x<1

Homework Equations

The Attempt at a Solution

We will use a proof by contrapositive.
We assume x>1 and we want to show x^2>x.
Let x be greater than 1 and let x^2=x*x.
If x^2=x*x, then x*x>x. Therefore, x^2>x.

 

[kidmode01]

Not quite, your contrapositive is correct and your first step assuming x > 1 is correct. But then you said:

Let x be greater than 1 and let x^2=x*x.
If x^2=x*x, then x*x>x. Therefore, x^2>x.

Firstly x^2 = x*x is just taken for granted so you don't need to put it in. Secondly, why does x^2=x*x imply x*x > x ?

All we know is:
x > 1
Then multiplying both sides by x we get:
x*x > 1*x
so then
x^2 > x

We can also prove this without the contrapositive.

 

[kathrynag]

Not quite, your contrapositive is correct and your first step assuming x > 1 is correct. But then you said:



Firstly x^2 = x*x is just taken for granted so you don't need to put it in. Secondly, why does x^2=x*x imply x*x > x ?

All we know is:
x > 1
Then multiplying both sides by x we get:
x*x > 1*x
so then
x^2 > x

.

So, is this all I would have to do - the multiplying by x?

 

[kidmode01]

Think about it for a little while. Every step has to follow logically from the previous and all we had to work with was a single assumption. Search the forum for similar inequality proofs or do a Google search on introduction to proofs. If you have a textbook with a section on proofs and the different kinds, that would also help looking at.

 

[Mark44]

Homework Statement

x^2<x, then x<1

Homework Equations

The Attempt at a Solution

We will use a proof by contrapositive.
We assume x>1 and we want to show x^2>x.
Let x be greater than 1 and let x^2=x*x.
If x^2=x*x, then x*x>x. Therefore, x^2>x.

As kidmode pointed out, you can do this directly. You also need to assume that x > 0, because the inequality isn't true otherwise. E.g., if x = -1/2, x^2 = 1/4 > -1/2, and if x = -2, x^2 = 4 > -2.

Start by assuming x > 0 and that x^2 < x.
Then x^2 - x < 0.
Now factor the left side and determine which values of x make it true.

 

[kidmode01]

Whoops. I wrote it down but I didn't add it, thanks Mark44.

 

[HallsofIvy]

I hate to add a complication but the negation of "x< 1" is NOT "x> 1".

 

[kidmode01]

Whoops again.

~(x<1) = (x >= 1)

Just put a line underneath all the inequalities in your contrapositive proof lol.
You'll have to fix the negation of the right side of "if" statement.
You'll end up proving:

x^2 >= x

Thanks HallsofIvy

 

[kathrynag]

Well, I'm supposed to prove using the contrapositive.

 

[kidmode01]

"P implies Q" and "Not Q implies Not P" are logically equivalent. Constructing a truth table will show that the truth values of both statements are the same. So proving the statement is true or it's contrapositive is true are equivalent.

 

[Mark44]

Well, I'm supposed to prove using the contrapositive.

That wasn't clear to me from your first post "We will use a proof by contrapositive."

I interpreted that to mean that this was the direction you had decided to go, not one that was mandated in the problem.