Operations on both sides of an equation

[Bipolarity]

What allows you to add the same quantity to both sides of an equation? Is this an axiom? A property of addition by which addition is defined? Or both? What about for multiplication?

If you differentiate both sides of an equation, the resulting equation will be valid, provided both sides of the original equation are differentiable.

On the other and, if you integrate both sides of an equation, the results will differ by some constant.

So some operations preserve uniqueness, some don't. How does one know that addition and multiplication yields a unique result? It's an axiom, right? Using these axioms, can we prove this property for subtraction and division?

Thanks!

BiP

 

[HallsofIvy]

It's pretty much part of the definition of "operation" that it be "well defined". That is if a= b then
f(a)= f(b). For example, the sum of two numbers is "well defined" because if a= b then a+ x= b+ x for any number x. Same for the product of two numbers.

 

[h6ss]

The set of real numbers ordered by the usual less than "<" or greater than ">" relations is totally ordered. From the strict order relation, we have that for x and y in ℝ, either x<y, x=y or x>y. Therefore the property of compatibility says that if x<y, then x+z<y+z and in particular, x-y<y-y=0; and if x<y and z>0, then xz<yz, for z also in ℝ. We would get equivalently equal and logical results for x=y and x<y.

I would consider them "fundamental properties".

 

[SteamKing]

Why wouldn't you be allowed to add the same quantity to both sides of an equation?

 

[Student100]

I don't think any time you alter the equation that uniqueness is preserved. 1=1 isn't equal to 1(1)=1(1), it's equivalent. Same as 1/2 and 4/8. Most times it doesn't matter unless you're dealing with functions.. Then you got to be extra careful.

But I think you're referring to the property of equality, if I'm reading the question right.

 

[HallsofIvy]

Have you never actually taken a math class? 1(1)= 1(1) and 1= 1 are exactly the same, they are just different ways of saying the same thing. If I say "it is snowing" and another person says "snow is falling" we are using different words but saying exactly the same thing.

 

[Bipolarity]

Have you never actually taken a math class? 1(1)= 1(1) and 1= 1 are exactly the same, they are just different ways of saying the same thing. If I say "it is snowing" and another person says "snow is falling" we are using different words but saying exactly the same thing.

Are you referring to me or to Student100?

BiP

 

[HallsofIvy]

I was responding to Student100.

 

[Mark44]

I don't think any time you alter the equation that uniqueness is preserved. 1=1 isn't equal to 1(1)=1(1), it's equivalent.

I don't understand what you're trying to say here. How is uniqueness important in equations? These equations are all equivalent:
x = 1
x + 1 = 2
2x - 2 = 0

Also, you can't say that a thing is equivalent ("it's equivalent"). There always has to be another thing around for the comparison.

Same as 1/2 and 4/8. Most times it doesn't matter unless you're dealing with functions.. Then you got to be extra careful.

But I think you're referring to the property of equality, if I'm reading the question right.

 

[Student100]

Ugh I was saying 1=1 is equivalent to 1(1)=(1)1 but each is its own unique equation. It isn't the same thing, that's idiotic. They're equivalent. Just the same as 1/2 and 4/8 are equivalent to each other... But not equal depending on the context. One out of two people... Or four out of eight people... It's not the same damn thing.

What I'm saying is that anytime you do an operation to both both sides of the equation you create an equation that's equivalent to the first, but that both equations are unique from one another. So the poster asked what operations preserve uniqueness so I would say none.

In your own example hell you're basically illustrating my point, the resulting information of the two conversations is basically the same but the words are unique. This is why you plug solutions into the original equation and not your simplified version or know the domain of the original.

 

[Student100]

Maybe this example will help illustrate what I mean.
You have two rooms with a ratio of people who have apples or don't, the rooms are equal.
f(a)=f(b)
a=1/2
a=b
1/2=1/2
Now we use the properties to add one to both sides..
1/2+1/1=1/2+1/1
Mathematically it's still true but the information it provides differs from the original equation. I now have more people with apples in each room than when I started. Each equation is unique.

I thought this is what the poster was saying when he referred to uniqueness. So using equality properties to alter an equation never creates an equation that preserves the uniqueness of the original.

Maybe I misunderstood what he was asking.

 

[austinuni]

What allows you to add the same quantity to both sides of an equation? Is this an axiom? A property of addition by which addition is defined? Or both? What about for multiplication?

In his book _Introduction to Logic and to the Methodology of Deductive Sciences_, Tarski gives a proof of this.
The theorem: if y=z, then x+y = x+z (adding x to both sides of the equation)
1) x=x from the law of identity
2) There exists a 'z' such that x+y = z
3) From (1) and (2), x+y = x+y
Then it's just a short step to the theorem.