# Logic

##### Full Member
x,y is variables, "=" is equal like math "=" ..nothing else.
what's struggling me is: yeah x=y right? but who will ensure for me that if I progress in my analysis and arrived to y=z, that's x=y still right? if so then we can say x=z ! but if I didn't change the equation x=y who ensure for me that it's really didn't change that equation and still right for any circumstances ?!

what's confusing me this:
(1) x=y ..so fine with that.
after 5minutes of thinking and analysing problem, I get
(2) y=z

then we conclude that x=z, but what's confusing me why it's that right? who ensures that the first equation is still right and none changed it?! that's what confusing me alot!!!

##### Full Member
When we say $$\displaystyle x = y$$,

we mean that x and y represent the same thing. They are just different names for the same thing. If you own only one dog and it is named Toto, it makes no difference whether you say "my dog" or "Toto" because both refer to the same animal.

By "same thing" in math, we frequently mean "have the same numerical value."

So I can say $$\displaystyle 13 + 3 = 21 - 5$$

because both expressions evaluate to the same result, namely 16.

You seem to be taking something that is easy and making it hard. In the case of 13 + 3 and 21 - 5, the expressions themselves are different, but the value of the expressions is identical. We look beyond the superficial form of the expressions and consider the quantitative meaning represented by the expressions.

So yes we can replace 13 + 3 by 21 - 5 or replace 21 - 5 by 13 + 3 because both expressions are quantitatively identical.
I got you !
but lemme ask something else, if I assign "my dog"=z , then we can call "my dog" as z?! what's confusing me, we must define "my dog" =: z and not "my dog" = z ! there's difference between "=" and "=:"(definition in math) .. but it seems the same ? I mean the mean of "=" is the same as "=:" ?!

##### Full Member
Are you talking about equality vs assignment?
well what's confusing me is how can I do x=y and then say that's y is the same as x, but I didn't before define what's y !

#### lev888

##### Full Member
Could you post an example of a problem that involves the issue that's confusing you? Otherwise it's very hard to understand you.

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#### JeffM

##### Elite Member
before doing for example x=y , musn't I define it before as x=y and then do in my solution's analysis x=y ?! I mean how can I assume that x=y which I didn't define before x=:y !

is "=" implicitly defining the variable that I'm using?!
I tried to avoid complexity in my original answer. The = sign has slightly different technical meanings in different contexts, but it always mean that two things can be treated as being the same for the purpose that is relevant.

In arithmetic and elementary algebra, we are interested in numeric value. So there the = sign means that two variables have the same numeric value. So if I know that x = y (meaning that they have the same numeric value), it makes no difference numerically which is used. It is really that simple.

Now how do I know that two expressions represent the same numeric value?

They may have been defined to do so as in

$$\displaystyle \text {Let } x = \dfrac{u^2 - 1}{v^2 + 1}.$$

You can use x in place of the fraction.

You may have proved that two values are necessarily the same. For example, you can prove that

$$\displaystyle a^2 + b^2 = c^2 \implies |c| = \sqrt{a^2 + b^2}.$$

Or it may be imposed as a condition of a problem. For example,

$$\displaystyle \text {Given } f = \dfrac{9c}{5} + 32, \text { where does } f = c.$$

So in this case, we are asking about a special case where f = c even though that is usually false.

$$\displaystyle \text {If } f = c, \text { then } = \dfrac{9c}{5} + 32 \implies f = \dfrac{9f}{5} + 32 \implies 5f = 9f + 160 \implies$$

$$\displaystyle 9f - 5f = -\ 160 \implies f = -\ 40 = c.$$

HOWEVER you come to know that x = y, you can THEREAFTER replace x with y or replace y by x. In particular if you know x = y and y = z, you can replace y in the second equation to get x = z.

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##### Full Member
Hi guys, I'm totally confused on that thing and by you guys I believe that I succeed to understand whole math problems (implicitly)

lets assume it's given like this : F(x)=1/(2*x +3)
so if I have arrived to an equation like 1 / (2*3y +3) , so here x=3y ! how is that right? I mean if it could be
1 / (2*y +3) then yeah I can say that x=y ! I'm fine with this, but how actually x=3y ? isn't the pattern of writing x in the prime equation say that we are "just" looking at x as concrete and not general? I mean cosmetics it's x, but x couldn't be represented as 2x also or 3x ? the matter that we are looking implicitly at x in general or what?! I mean in general which x could be 2x also or 4x or 10000000000000000000000000000x and it's still called "x" that we can assign it in the equation instead of prime x?! if yes, then why? it's eyes like something not identical to x, I mean 1000000000000000000x isn't identical to x ?!

#### pka

##### Elite Member
lets assume it's given like this : $$\displaystyle F(x)=\frac{1}{2x +3}$$
Can you find the value of ? if $$\displaystyle F(?)=\frac{1}{4t^2 +9}$$.