Contrapositive Proof of Positive x & y: x^n<y^n implies x<y

In summary, the conversation discussed using induction to prove that if both x and y are positive, then x<y implies x^n<y^n. The second part focused on proving the converse, that if both x and y are positive, then x^n<y^n implies x<y, using the indirect proof using contrapositive. It was mentioned that the theorem only applies to positive numbers and that the contrapositive requires the hypothesis and conclusion to be negated and reversed. The formula for factoring x^n- y^n was also discussed.
  • #1
garyljc
103
0

Homework Statement


1st part , using induction to prvoe that if both x and y are positive then x<y implies x^n<y^n
2nd part, prove the converse, that if both x and y are postive then x^n<y^n implies x<y

Homework Equations


my question is more on the second part. I understand that I have to use the indirect proof using contrapositive.


The Attempt at a Solution


I started with this
If x and y are not positive then it does not imply the following x^n<y^n implies x<y
 
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  • #2
garyljc said:

Homework Statement


1st part , using induction to prvoe that if both x and y are positive then x<y implies x^n<y^n
2nd part, prove the converse, that if both x and y are postive then x^n<y^n implies x<y

Homework Equations


my question is more on the second part. I understand that I have to use the indirect proof using contrapositive.


The Attempt at a Solution


I started with this
If x and y are not positive then it does not imply the following x^n<y^n implies x<y
First, what you have written here is irrelevant. The theorem (and therefore its contrapositive) talks only about positive numbers. "x and y both positive" is not part of the hypothesis, it is a "preliminary requirement" What is true if x and y are not positive does not matter. The theorem itself is simply "x^n< y^n implies x< y".

Are you clear on what the contrapositive is here? The contrapositive requires that the hypothesis and conclusion be negated and reversed. "x and y both positive" is not part of the hypothesis, it is a "preliminary requirement" that stays the same.

If x and y are positive numbers, then [itex]x^n\ge y^n[/itex] implies [itex]x\ge y[/itex].

Also, while you can prove it using induction, you don't have to. [itex]x^n
\ge y^n[/itex] implies that [itex]x^n- y^n\ge 0[/itex] and [itex]x^n- y^n= (x- y)(x^{n-1}+ x^{n-2}y+ \cdot\cdot\cdot+ xy^{n-2}+ y^{n-2}[/itex]
 
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  • #3
i do understand contrapositive
but I'm new to it
does it mean that I'm suppose to prove if x^n < y^n is false therefore it implies that x < y is false ?
am i heading the correct direction ?
 
  • #4
HallsofIvy said:
Also, while you can prove it using induction, you don't have to. [itex]x^n
ge y^n[/itex] implies that [itex]x^n- y^n> 0[/itex] and [itex]x^n- y^n= (x- y)(x^{n-1}+ x^{n-2}y+ \cdot\cdot\cdot+ xy^{n-2}+ y^{n-2}[/itex]

i do not understand how did you arrive at this conclusion
 
  • #5
garyljc said:
i do understand contrapositive
but I'm new to it
does it mean that I'm suppose to prove if x^n < y^n is false therefore it implies that x < y is false ?
am i heading the correct direction ?
Yes, you are. Don't forget that you are assuming also that x and y are positive and that "x^n< y^n is false" is the same as "[itex]x^n\ge y^n[/itex]".

garyljc said:
i do not understand how did you arrive at this conclusion
What conclusion do you mean? If you are talking about the factoring (which is not a "conclusion"), what do you get if you multiply (x- y) and (xn-1+ yxn-2+ ...+ xyn-2+ yn-1? And what would it say if n= 2?
 
  • #6
What conclusion do you mean? If you are talking about the factoring (which is not a "conclusion"), what do you get if you multiply (x- y) and (xn-1+ yxn-2+ ...+ xyn-2+ yn-1? And what would it say if n= 2?[/QUOTE]


sorry i meant like who did you factor that up ?
is there a formula to help me with ?
 
  • #7
Did you kow that x2- y2= (x- y)(x+ y)? How about x3- y3= (x- y)(x2+ xy+ y2)?

Yes, there is a formula: it is exactly what I gave: xn- yn= (x- y)(xn-1+ xn-2y+ ...+ xyn-2+ yn-1. And, again, you can prove that by multiply the right hand side.
 
  • #8
Thanks . Got it =)
 

What is a contrapositive proof?

A contrapositive proof is a type of mathematical proof used to demonstrate the validity of a statement. It involves proving the statement's contrapositive form, which is the statement formed by negating both the hypothesis and conclusion.

How is a contrapositive proof different from a direct proof?

Unlike a direct proof, which involves directly proving the statement's hypothesis and conclusion, a contrapositive proof involves proving the statement's contrapositive form. This can be helpful in cases where directly proving the statement is difficult or impossible.

What is the statement being proven in a contrapositive proof of positive x & y: x^n

The statement being proven is that if x and y are positive real numbers and x raised to the power of n is less than y raised to the power of n, then x is less than y.

Why is it necessary to assume that x and y are positive in this statement?

It is necessary to assume that x and y are positive because the statement being proven involves raising these numbers to a power. If x and y were negative, the statement would not be true. Additionally, assuming that x and y are positive makes the statement more specific and easier to prove.

What are some real-world applications of contrapositive proofs?

Contrapositive proofs are commonly used in fields such as computer science, engineering, and physics to prove theorems and propositions. They can also be applied in decision-making, problem-solving, and logic puzzles.

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