# Thread: what allow us to put x=a+b+c

1. ## what allow us to put x=a+b+c

Many times in mathematics in proving something we can put

x=a+b+c or x= a-b^2 or x=y e.t.c.

What axiom or theorem in mathematics allow us to do that??

2. Originally Posted by triclino
What axiom or theorem in mathematics allow us to do that??
None. We are just using what the problem is saying. Say that $\displaystyle f: \mathbb{R}\to \mathbb{R}$ is such a function so that $\displaystyle f(x) = f(\tfrac{x}{2})$. This means $\displaystyle f(2y) = f(y)$ for any $\displaystyle y\in \mathbb{R}$. Thus, $\displaystyle f$ has property that $\displaystyle f(x) = f(2x)$ for any $\displaystyle x$.
This is just using the definition of what the function is.

3. Originally Posted by ThePerfectHacker
None. We are just using what the problem is saying. Say that $\displaystyle f: \mathbb{R}\to \mathbb{R}$ is such a function so that $\displaystyle f(x) = f(\tfrac{x}{2})$. This means $\displaystyle f(2y) = f(y)$ for any $\displaystyle y\in \mathbb{R}$. Thus, $\displaystyle f$ has property that $\displaystyle f(x) = f(2x)$ for any $\displaystyle x$.
This is just using the definition of what the function is.
Very interesting your function concept,but then again what allow us for the creation of a function.

in the following example how the function concept could be used?

suppose we want to prove that:

...................(a+b+c)/3 >= (abc)^1/3,with a,b,c>=0

Now without using the Am-Gm concept for the proof of the inequality,
one way of solving the inequality is to use the substitution

......................x=(a+b+c)/3

4. Originally Posted by triclino
Very interesting your function concept,but then again what allow us for the creation of a function.
There is no theorem that tells us we can do that. It is just by definition. If $\displaystyle y$ is any real number then $\displaystyle 2y$ is a real number and so $\displaystyle f(2y)=f(y)$. That is just the definition.

5. Originally Posted by ThePerfectHacker
There is no theorem that tells us we can do that. It is just by definition. If $\displaystyle y$ is any real number then $\displaystyle 2y$ is a real number and so $\displaystyle f(2y)=f(y)$. That is just the definition.
The axiom that allow us for the creation of a function is the axiom of

..........specification in set theory............................................ .....

6. Originally Posted by triclino
The axiom that allow us for the creation of a function is the axiom of
There is NO axiom.
This is what I told you the first two times.
Just look at what was done.

7. Originally Posted by triclino
Very interesting your function concept,but then again what allow us for the creation of a function.

in the following example how the function concept could be used?

suppose we want to prove that:

...................(a+b+c)/3 >= (abc)^1/3,with a,b,c>=0

Now without using the Am-Gm concept for the proof of the inequality,
one way of solving the inequality is to use the substitution

......................x=(a+b+c)/3
I would like to give a nice proof for this inequality:

$\displaystyle (abc)^{1/3}$

$\displaystyle =\exp\big\{\ln\big((abc)^{1/3}\big)\big\}$

$\displaystyle =\exp\bigg\{\frac{1}{3}\ln(a)+\frac{1}{3}\ln(b)+\f rac{1}{3}\ln(c)\bigg\}$

$\displaystyle \leq\frac{1}{3}\exp\big\{\ln(a)\big\}+\frac{1}{3}\ exp\big\{\ln(b)\big\}+\frac{1}{3}\exp\big\{\ln(c)\ big\}\text{ convex function property}$

$\displaystyle =\frac{1}{3}\big(a+b+c\big),$

which is the desired result (also see the proof for Young's inequality).

As ThePerfectHacker mentioned, there is no axiom on the things you told.
You just find your way yourself by picking new parameters or arranging the given ones to reveal your own way.
Ofcourse, experience is the most important thing in this dark forest!

8. In my very 1st post i asked:

......what allows us to put x=a+b+c ...............................................

in a proof that uses this substitution.And then i brought up as an example to the posts of the PerfectHacker the said inequality

Anyway in your proof you do not use ,if i am not mistaken,any substitutions

PerfectHacker brought in the concept of function and i was forced to point out that even the concept of function needs a theorem of existence.

So now we are faced with two questions ;

1)If the concept of function needs a theorem of existence,or

2)The substitution x=a+b+c, or any substitution of this type need a theorem or axiom to back it up

For example the concept of sqroot need a theorem of existence before it is defined

Notice, my ' or" is not exclusively disjunctive

9. I am sorry, but i forgot to say that in step wise proof ( = formal) where each step is justified ,when a substitution of this type is done,( x=a+b+c or x=a+b or e.t.c e.t.c ) one will need to justified that

10. Originally Posted by triclino
PerfectHacker brought in the concept of function and i was forced to point out that even the concept of function needs a theorem of existence.
Are you asking why we know there is such a function $\displaystyle f:\mathbb{R}\to \mathbb{R}$ such that $\displaystyle f(x) = f(\tfrac{x}{2})$? Is that why you mean by existence? If so it does not matter - that was simply for an example. We can assume it exists. And see where we get from there.

11. Originally Posted by ThePerfectHacker
Are you asking why we know there is such a function $\displaystyle f:\mathbb{R}\to \mathbb{R}$ such that $\displaystyle f(x) = f(\tfrac{x}{2})$? Is that why you mean by existence? If so it does not matter - that was simply for an example. We can assume it exists. And see where we get from there.
As i said in a step wise proof, that everything is justified we cannot assume anything to exist unless you do not accept formal proofs

12. Originally Posted by triclino
As i said in a step wise proof, that everything is justified we cannot assume anything to exist unless you do not accept formal proofs
Yes you can assume. In math we always make assumptions.

However, if a proof depends on the assumption it is not a proof. You need to prove the assumption.

In the example I gave you there is nothing bad. I was not trying to prove anything. Just give you an example. Everything was justified.