Proving Inequality using Arithmetic and Order Axioms

In summary: From there, you can use the distributive law to combine the two inequalities, and ultimately show that ##x\geq y##.
  • #1
synkk
216
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Using only the axioms of arithmetic and order, show that:

for all x,y satisfy 0≤x, 0≤y and x≤y, then x.x ≤ y.y

I'm really lost on where to start, my attempt so far was this

as 0 <= x and 0 <= y, we have 0 <= xy from axiom (for all x,y,z x<=y and 0<=z, then x.z <=y.z). then we use the same axiom to get y.y >= 0, but not sure where to go from there
 
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  • #2
synkk said:
Using only the axioms of arithmetic and order, show that:

for all x,y satisfy 0≤x, 0≤y and x≤y, then x.x ≤ y.y

I'm really lost on where to start, my attempt so far was this

as 0 <= x and 0 <= y, we have 0 <= xy from axiom (for all x,y,z x<=y and 0<=z, then x.z <=y.z). then we use the same axiom to get y.y >= 0, but not sure where to go from there

There are several versions of the "axioms", so you need to tell us exactly what axioms you are given.
 
  • #3
synkk said:
Using only the axioms of arithmetic and order, show that:

for all x,y satisfy 0≤x, 0≤y and x≤y, then x.x ≤ y.y

I'm really lost on where to start, my attempt so far was this

as 0 <= x and 0 <= y, we have 0 <= xy from axiom (for all x,y,z x<=y and 0<=z, then x.z <=y.z). then we use the same axiom to get y.y >= 0, but not sure where to go from there

Do you see how your axiom, when applied to the hypothesis that 0≤x, 0≤y and x≤y, gives you both x.x≤y.x and x.y≤y.y?

Do you see where to go from here?
 
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  • #4
Thanks I got it from there.

How would I go about proving it the other way? I.e. if 0 <= x, 0 <= y, x.x <= y.y then x <= y? How would I prove that? I have gotten to (x-y).(x-y) <= 0 but not sure how to actually prove it using the axioms which are:

A1 (x+y)+z = x+(y+z)
A2 x + y = y + x
A3 x + 0 = x for all x
A3 x + (-x) = 0
A5 x.(y.z) = (x.y).z
A6 x.y = y.x
A7 x.1 = x
A8 x.x^(-1) = 1
A9 x.(y+z) = (x.y) +(x.z)

A10 x<=y or y<=x for all x,y
A11 if x<=y and y <=x then x = y
A12 x<=y and y <=z then x<=z
A13 x<=y then x + z <= y + z
A14 x<=y and 0<=z then x.z <=y.z
 
  • #5
synkk said:
Thanks I got it from there.

How would I go about proving it the other way? I.e. if 0 <= x, 0 <= y, x.x <= y.y then x <= y? How would I prove that?

I think the easiest way to prove this is by contradiction.

I have gotten to (x-y).(x-y) <= 0 but not sure how to actually prove it using the axioms which are:

...

If you want to try this, I think it'd be easier to start with ##0\leq y^2-x^2=(y+x)(y-x)##.
 
Last edited:

1. What are arithmetic and order axioms?

Arithmetic and order axioms are fundamental principles in mathematics that allow us to perform operations and make comparisons between numbers. They provide the basic rules for addition, subtraction, multiplication, and division, as well as the properties of equality and inequality.

2. How can arithmetic and order axioms be used to prove inequalities?

Arithmetic and order axioms can be used to prove inequalities by applying them to specific numerical expressions or equations. By using these axioms, we can manipulate the expressions or equations to show that one side is greater than the other, thereby proving the inequality.

3. What are some common examples of inequalities that can be proven using arithmetic and order axioms?

Some common examples of inequalities that can be proven using arithmetic and order axioms include greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). These inequalities are often used in simple equations or in more complex mathematical proofs.

4. Can arithmetic and order axioms be applied to all types of numbers?

Yes, arithmetic and order axioms can be applied to all types of numbers, including integers, fractions, decimals, and even irrational numbers. These axioms provide a universal set of rules that can be used to manipulate and compare numbers of any type.

5. How can understanding arithmetic and order axioms help in solving real-world problems?

Understanding arithmetic and order axioms can help in solving real-world problems by providing a logical and systematic approach to solving mathematical equations and inequalities. By using these axioms, we can accurately compare and manipulate different quantities, making it easier to solve problems in fields such as finance, engineering, and science.

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