# 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!!!

#### Ryan$##### Full Member if I didn't change the equation (1), why would it still right?! who ensure that none touch it and change the equation? I mean yeah x=y, but I went to other calculations while thinking and arrived to equation (2) ! but once again who said that equation (1) still satisfying, yeah I didn't change it ..but who said if I didn't change it, then it's still right for other cases/circumstances I got in progress of my solution?! #### pka ##### Elite Member Do you think that the transitive relationship is valid? If so apply it here. If not, why not? • topsquark #### JeffM ##### Elite Member if I didn't change the equation (1), why would it still right?! who ensure that none touch it and change the equation? I mean yeah x=y, but I went to other calculations while thinking and arrived to equation (2) ! but once again who said that equation (1) still satisfying, yeah I didn't change it ..but who said if I didn't change it, then it's still right for other cases/circumstances I got in progress of my solution?! In the context of a specific problem, the meanings of x, y, and z do not change. So if x is the same as y, and z is the same as y, and (for purposes of a given problem) the meanings of x, y, and z do not change, then x and z are also the same. Remember that variables stand for numbers. $$\displaystyle 10 + 7 = 20 - 3.$$ $$\displaystyle 33 - 16 = 20 - 3$$ So is it true or false that $$\displaystyle 10 + 7 = 33 - 16.$$ The statement $$\displaystyle x = y \text { and } z = y \implies x = z$$ is merely a generalization of that example. • topsquark #### Otis ##### Senior Member ... my question in the book given that x=z, why? Is that book written in English? (Maybe that's part of your issue.) If you and I have exactly$1 in our pocket, and somebody else says to you, "I have exactly $1 in my pocket, just like you", then why think the$1 in my pocket would suddenly change. Magic? ##### Full 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?!

#### lev888

##### Full Member
Are you talking about equality vs assignment?

• • Ryan$and topsquark #### Ryan$

##### 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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• topsquark and Ryan$#### Ryan$

##### Full Member
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.

I'm totally with you, but then what's the purpose of "=:" which it's for definitions ! ?! I can say x =: y is the same as x=y..what's wrong with?

#### JeffM

##### Elite Member
What's wrong is that

$$\displaystyle x =: y \implies x = y$$, but

$$\displaystyle x = y \not \implies x = y:.$$

Remember that I said there were a number of different situation where it makes sense to say x = y. Only one of those situations involves definition.

Now I agree that in many cases people mean

$$\displaystyle x \equiv y$$ when they say $$\displaystyle x = y.$$

Because $$\displaystyle x \equiv y \implies x = y$$,

that informal usage seldom causes any harm whatsoever.

I must admit that you seem to be doing your best to create confusion where none need exist.

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• MarkFL and topsquark

#### Ryan\$

##### 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 $$\displaystyle F(?)=\frac{1}{4t^2 +9}$$.