# how is possibill demonstring ?

#### jhonyy9

##### New member
- let n greater or equal 3 from the set of real numbers R,
- let a and b graters or equal 1 from the set of real numbers R,
- we have the equation : n=a+b+1

- what will be one demonstration that for every value of number n always will be,always there two numbers a and b for this equation is true ?

Thank you.

#### jhonyy9

##### New member
- if n is a number greater or equal 3 from the set of real numbers R,
- a and b are graters or equals with 1 from R,
- and we have the equation n=a+b+1

- what we have to demonstrate is that always,for every value of n will be one number a and one number b so that the equation will be true

#### jhonyy9

##### New member
- if n is greater or equal 3 ,we can writing n-1 greater or equal with 2
- if a and b are graters or equals with 1 ,we can writing that a+b is greater or equal with 2
- from this results that n-1=a+b ie where n=a+b+1

- this demonstration may be accepted ?

J

#### JeffM

##### Guest
PROVE
If n is a real number greater than or equal to 3,
Then there exists at least one pair of real numbers, a and b, each greater than or equal to 1, such that (a + b + 1) = n.

Is that the problem?
How have real numbers been defined for you? What can you assume?

IF YOU ARE IN BEGINNING ALGEBRA I see nothing wrong with a proof like:
Let a = n - 2.
So, a is a real number and a >= (3 - 2).
So, a is a real number >= 1.
Let b = 1.
So, b is a real number >= 1.
So, (a + b + 1) = (a + 1 + 1) = (a + 2) = (n - 2) + 2 = n.
(n - 2) and 1 are a pair of real numbers that meet the requirements.
So there is at least one such pair of real numbers.

IF YOU ARE IN SOME ADVANCED MATH COURSE AND MUST PROVE THAT THE REAL NUMBERS ARE CLOSED UNDER ADDITION, YOU ARE WAY BEYOND ME AND NEED TO POST THIS PROBLEM UNDER ADVANCED MATH.

#### jhonyy9

##### New member
- this equation can be demonstrated with complete mathematical induction ?
- if so,how it can be demonstrated ?

J

#### JeffM

##### Guest
jhonyy9 said:
- this equation can be demonstrated with complete mathematical induction ?
- if so,how it can be demonstrated ?
I gave you a constructive proof of what I THOUGHT was your problem. Clearly, I did not understand it. Sorry to be of no help.

PS. As you have defined the problem SO FAR, n is to be a REAL number >= 3. The proof I gave was for any such real number. There is no need for mathematical induction in the problem as posed.

#### Subhotosh Khan

##### Super Moderator
Staff member
jhonyy9 said:
- let n greater or equal 3 from the set of real numbers R,
- let a and b graters or equal 1 from the set of real numbers R,
- we have the equation : n=a+b+1

- what will be one demonstration that for every value of number n always will be,always there two numbers a and b for this equation is true ?
n= a+b+1 is true for n=3

assume that it is true for an arbitrary 'n'.

n = p + q + 1

Then:

n+1 = (p+q+1) + 1 = p + (q+1) + 1 = a + b + 1

This is what JeffM had presented. It is surprising that you could not rewrite that in this form!!!

#### lookagain

##### Senior Member
JeffM said:
IF YOU ARE IN SOME ADVANCED MATH COURSE AND MUST PROVE THAT THE REAL NUMBERS ARE CLOSED UNDER ADDITION, YOU ARE WAY BEYOND ME AND NEED TO POST THIS PROBLEM UNDER ADVANCED MATH.
You can state your message without yelling it by not using all caps, such as the above.
I have been corrected on this same technique before.

#### Denis

##### Senior Member
lookagain said:
JeffM said:
IF YOU ARE IN SOME ADVANCED MATH COURSE AND MUST PROVE THAT THE REAL NUMBERS ARE CLOSED UNDER ADDITION, YOU ARE WAY BEYOND ME AND NEED TO POST THIS PROBLEM UNDER ADVANCED MATH.
You can state your message without yelling it by not using all caps, such as the above.
And the all caps also shows a poor response to your impatience and frustration with
the user.
Sir, please let the moderators do any reprimanding.
It is the reason why they are moderators.
We thank you for your cooperation.

#### lookagain

##### Senior Member
Subhotosh Khan said:
n= a+b+1 is true for n=3

assume that it is true for an arbitrary 'n'.

n = p + q + 1

Then:

n+1 = (p+q+1) + 1 = p + (q+1) + 1 = $$\displaystyle \text{ Error here:}$$a + b + 1
$$\displaystyle n = p + q + 1$$

$$\displaystyle \text{Therefore}$$

$$\displaystyle n + 1 =$$

$$\displaystyle (p + q + 1) + 1 =$$

$$\displaystyle p + (q + 1) + 1 =$$

$$\displaystyle \text{What did you intend for your conclusion?}$$

#### Subhotosh Khan

##### Super Moderator
Staff member
lookagain said:
Subhotosh Khan said:
n= a+b+1 is true for n=3

assume that it is true for an arbitrary 'n'.

n = p + q + 1

Then:

n+1 = (p+q+1) + 1 = p + (q+1) + 1 = $$\displaystyle \text{ Error here:}$$a + b + 1
$$\displaystyle n = p + q + 1$$

$$\displaystyle \text{Therefore}$$

$$\displaystyle n + 1 =$$

$$\displaystyle (p + q + 1) + 1 =$$

$$\displaystyle p + (q + 1) + 1 =$$

$$\displaystyle \text{What did you intend for your conclusion?}$$

a = p
b = q+1

thus

n+1 = a + b + 1

Thus if (n+1) can be expressed as a+b+1.

Thus anunumber can be expressed as a+b+1

#### lookagain

##### Senior Member
Subhotosh Khan said:
a = p
b = q+1

thus

n+1 = a + b + 1

Thus if (n+1) can be expressed as a+b+1.

Thus anunumber can be expressed as a+b+1$$\displaystyle You \ can't \ use \ the \ same \ variables$$
$$\displaystyle here \ as \ were \ given \ in \ the \ problem.$$
But (n + 1) cannot be expressed as (a + b + 1), because n already equals (a + b + 1).

You have to use at least one different variable (~ letter) for your point.

n and (n + 1) cannot both be equal to (a + b + 1).

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#### Subhotosh Khan

##### Super Moderator
Staff member
Thank you for the correction.

#### jhonyy9

##### New member
- this equation can be demonstrated with complete mathematical induction ?

- n=a+b+1 --- 3=1+1+1 --- is true
--- 4=2+1+1 --- is true
--- 5=2+2+1 --- is true
- if n=m --- m=a+b+1 --- suppose is true,
then m+1=a+b+1+1
- m+1=a+1+b+1 --- is true -- commutativity
- m+1=a+(b+1)+1 --- is true --- associativity

- results that this equation n=a+b+1 -- is true

- this demonstration may be accepted ?

J

#### JeffM

##### Guest
jhonny9

In your first post, you said "let n greater or equal 3 from the set of real numbers R."

The proof you give in your last post is of a type that is valid for integers. If it is true for some integer k, it is true for the integer (k + 1). It is true for some integer m. Therefore it is true for any integer n $$\displaystyle \geq \$$ m. That precise form of proof is NOT valid for real numbers.

If we understand what you are being asked to prove (please see my first post under JeffM), it can be proved for any real number WITHOUT mathematical induction. If you are required to prove the statement for a real number using some form of mathematical induction, please let us know. I am sure that someone other than me can show you how that is done. We need you to tell us what is to be proved and by what method.

Again a valid CONSTRUCTIVE proof (treating as axiomatic that the system of real numbers is closed under addition) of what I THINK you are being asked is:

Let n be an ARBITRARY real number $$\displaystyle \geq \$$ 3.
Let a = (n - 2).
So a is a real number $$\displaystyle \geq \$$1.
Let b = 1.
So, b is a real number $$\displaystyle \geq \$$ 1.
(a + b + 1) = (a + 1 + 1) = (a + 2) = [(n - 2) + 2] = n.
So there is at least one pair of real numbers a and b such that a $$\displaystyle \geq \$$ 1 $$\displaystyle \leq \$$ b and (a + b + 1) = n for any real n $$\displaystyle \geq \$$ 3.

#### jhonyy9

##### New member
... this demonstration is like wrong ... or why dont reply me nobody nothing ???

#### jhonyy9

##### New member
sorry but i now i have got that this my problem have two pages ... so this demonstration with math complete induction what i have wrote on below , on end of the first page write me please this demonstration it is acceptable or ... what is your opinion,please ?

for JefM what i have asked is how is possible writing a demonstration that for every value of n,when n is greater or equal 3 from R will be always one number a and one number b , every two graters or equals 1 from R,so that the equation n=a+b+1 is true
- for example : 3=1+1+1 , 4=2+1+1 , 5=2+2+1 ,...

thank you very much !

J

#### JeffM

##### Guest
jhonyy9 said:
- this equation can be demonstrated with complete mathematical induction ?

- n=a+b+1 --- 3=1+1+1 --- is true
--- 4=2+1+1 --- is true
--- 5=2+2+1 --- is true
- if n=m --- m=a+b+1 --- suppose is true,
then m+1=a+b+1+1
- m+1=a+1+b+1 --- is true -- commutativity
- m+1=a+(b+1)+1 --- is true --- associativity

- results that this equation n=a+b+1 -- is true

- this demonstration may be accepted ?
In my opinion, your demonstration is not acceptable. You are effectively assuming that n is an integer, but you said that n is a real number.

As I have said twice now, there is a demonstration for any real n $$\displaystyle \geq\$$ 3 that does not require induction in its proof. Are you required to give a proof by induction?

#### jhonyy9

##### New member
- sorry - yes this is true ... - so how you think this ,please,will be true if a,b,n are numbers from the set of natural numbers N ?
- in this conditions - if a,b,n are from N - this prove can may be accepted ?