Meaning of an Equation

Question

What exactly is the meaning of the variables in an equation? That’s probably not a good way to write the question I’m trying to ask, but take the equation

$$x^2 = 4$$

Does the $x$ represent all values in the domain, and is the equation is asking us to find which values of $x$ the equation is true for?

Or is $x^2 = 4$ always true, and are we just looking for what values $x$ represents.

Answer 1

The equal sign has (at least) two meanings. In math we get the meaning from context. In computer science, the variations in meaning can be denoted "=", or ":=" or some such.

One meaning is the "equals of identity." If I write

$$\sin^2x +\cos^2 x = 1$$

I mean that it's true for every value of $x$.

The other meaning is "conditional equality". The equation is a condition and we want to know when it is true and when it isn't.

$$x^2 = 4$$

is a conditional equality. It's true only under the condition that $x =2$ or $-2$.

Answer 2

In the context you presented:

You're looking for the values of $x$ that make the equation true. It's not going to be true for all $x$ (take $x=1$). You could view this as finding the intersections of the graphs of the left and right sides. That is, you can plot the left-hand side and the right-hand side, and solutions (values of $x$ that your equation true) correspond to where these lines intersect. In your example, you'll find that these lines intersect at two points, $(2,4)$ and $(-2,4)$. This contrasts with the concept of a function, where we assign, to each value of $x$, another number $f(x).$ As an example, take $f(x)=x^2-4.$ This is an assignment of a value of $x$ to another number $x^2-4.$ Note that the values where this is zero are exactly the values of $x$ that satisfy $x^2=4.$ The distinction between a function and an equation is an important one to make (and can sometimes become more confusing when you introduce implicit equations), but they can be related in the manner I did above. You can also have equality of functions, in which case you have $f(x)=g(x)$ for all $x$. The other post by M. Goddard presents an example of something like that ($\sin^2x+\cos^2x=1$). Discernment between all of these cases can be gained from context, which takes experience!

Answer 3

Short answer :

• "x²= 4" is a conditional equation : "if x= ..... then x²= 4"

• " x+y = y+x " is an identity ( unconditionnal equation, true for all possible values of x and y )

• in " the set of all x such that x²=4 " the expression " x²=4 " is a rule or a "law" that defines an object, here a set. Same thing in : " the curve ( set of points (x,y) such that x²=4 " ( that is a parabola with vertex (-4, 0) )

---

As such an expression like this : "x² = 4" is an open sentence. An open sentence as such is not a proposition, it has no truth value. The reason why it does not have a truth vale ( true or false) is that it does not have a complete meaning.

You can produce a true or false statement by completing its meaning in different ways ( using quantification ) .

• For all x ( belonging to R) : x² = 4 ( FALSE)

• The exists an x ( belonging to R) such that : x² = 4. ( True)

• For all x ( belonging to R) : [ x² = 4 iff (x = 2 OR x = -2) ] . ( True)

• For all x ( belonging to R) : x² = 4 iff x = 2 . ( False)

• For all x (belonging to R) : if x = 2 then x² = 4 . ( True)

• For all x ( belonging to R) : x² is not equal to 4. ( False)

When we solve an equation, the implicit context is :

           There is (at least) an x such that : x² = 4. 

This is not an open sentence: it is a complete sentence and a true one.

Your job is now to find which possible values of the unknown x makes the statement true.

Note : The truth of " there is at least an x such that..." is only presupposed, but sometimes we discover that this presupposition was wrong. Suppose I am given the equation :

                               x+1 = x 

I will reason like this :

(1) if x+1 = x then (x+1) - x = x - x

(2) if (x+1) - x = x - x then 1 = O

(3) but it is false that : 1=0

(4) therefore : It is false that there is an x such that : x+1 = x

Some open sentences, called " identities, are true for all the possible values of their variables. This is the reason why identities are often stated without quantifiers. But rigorously we should write :

     For all a & for all b ( belonging to R) : (a+b)² = a² + b²+ 2ab. 

To come closer to you question, I think it can be understood in the following way : is x an unknown or a variable?

If the expression " x² = 2 " is considered as an open sentence, x is an unknown, and the question is : what values of x have to be substituted for x to make the open sentence true?

But sometimes, the expression " x²= 4 " can be considered as a sort of "law" defining a set, a law that is supposed to be " always true" for all the elements of a set, and your job is to find which set corresponds to this law. This set can be a set of numbers , or a set of ordered pairs ( a relation, even a function).

For example, I can define a set like this.

S = { x belonging to R | x² = 4 }

The "law" : x² = 4 is , by definition, true of all the elements of S= { -2, 2 }.

I can also say, in coordinate geometry :

T = { (x, y) | x² = 4 }

T is the set of all points ( x,y) such that ( whatever y might be) : x²=4.

You can use a graphing calculator to see which curve this law ( x² = 4 ) defines.

When x is considered as a "law" or a defining formula ( for a set, a relation, a function) x is considered as a variable ( not as an unknown).

Note : you might ask, what about the alledged "law" : x+1 = x. Is it really a law defining an object, since there is no x such that : x+1 = x.

In fact, this law defines an object that is simply : the Empty Set

         { x | x+1 = x }= {    }