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Axioms for the real numbers.

paulmdrdo

Active member
May 13, 2013
386
in this problem we drop the use of parentheses when this step is justified by associative axioms. thus we write $\displaystyle x^2+2x+3\,\,instead\,\,of\,\,\left(x^2+2x\right)+3\,or\,x^2+\left(2x+3\right)$. tell what axioms justify the statement:

1. $\displaystyle \left(x^2+2x+5\right)+\left(x^2+3x+1\right)\,=\, \left(1+1\right)x^2+\left(2+3\right)x+ \left(5+1\right)$

i don't understand the question.
 
Last edited:

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,123
in this problem we drop the use of parentheses when this step is justified by associative axioms. thus we write $\displaystyle x^2+2x+3\,\,instead\,\,of\,\,\left(x^2+2x\right)+3\,or\,x^2+\left(2x+3\right)$. tell what axioms justify the statement:

1. $\displaystyle \left(x^2+2x+5\right)+\left(x^2+3x+1\right)\,=\, \left(1+1\right)x^2+\left(2+3\right)x+ \left(5+1\right)$

i don't understand the question.
You first have to prove: (a + b) + c = a + (b + c) = a + b + c. (I'm assuming the final form is meant to suggest addition of the terms in any order.)

Then for problem 1 use the above result to remove the parenthesis, use commutivity of addition to rearrange the terms, then use the distributive property to factor.

-Dan
 
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paulmdrdo

Active member
May 13, 2013
386
why did you use associativity of addition?
 

paulmdrdo

Active member
May 13, 2013
386
i still don't understand what the question means.
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,123
i still don't understand what the question means.
I'm assuming that if the addition is associative and commutative then we can show
(a + b) + c = (a + c) + b = (b + c) + a .... = a + b + c because we can show that order doesn't matter. So we simply call it a + b + c.

The problem is asking you to use this to remove the parenthesis in the following:
[tex](x^2 + 2x + 5) + (x^2 + 3x + 1) = x^2 + 2x + 5 + x^2 + 3x + 1[/tex]

To get to the final form you can use commutivity to rearrange the terms, then use the distributive property to factor them to the final form.

-Dan
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
tell what axioms justify the statement:

1. $\displaystyle \left(x^2+2x+5\right)+\left(x^2+3x+1\right)\,=\, \left(1+1\right)x^2+\left(2+3\right)x+ \left(5+1\right)$

i don't understand the question.
The answer to this question should be a list of axioms. The axioms in question are used in a proof of the equality above. Roughly speaking, a proof in this case is a chain of expressions $E_1=E_2=\dots=E_n$ where each $E_i$ has some subexpression $e$, $E_{i+1}$ is obtained from $E_i$ by replacing $e$ with $e'$ and $e=e'$ or $e'=e$ is an instance of an axiom of real numbers. For example, a proof may start with \[(x^2 + 2x + 5) + (x^2 + 3x + 1)=(1\cdot x^2 + 2x + 5) + (x^2 + 3x + 1)\]Here $E_1$ is $(x^2 + 2x + 5) + (x^2 + 3x + 1)$, $e$ is $x^2$ and $e'$ is $1\cdot x^2$. The axiom used here is $1\cdot x=x$ for all $x$, and $1\cdot x^2=x^2$ is its instance.

So you need to list all axioms that are used in the chain of equalities \[(x^2+2x+5)+(x^2+3x+1)=\dots=(1+1)x^2+(2+3)x+ (5+1)\]