The Difference Between Functions And Equations?

[ScienceNerd36]

Hello there my fellow complex number comrades,

I was just wondering, what are the key differences between a function and an equation?

 

[Office_Shredder]

Depends on how technical you want to be. What you're probably looking for is something like this:

A function is usually described by an equation f(x)=...*insert what f(x) actually is* or through a similar method. The formal definition of a function is a set of pairs {(x,y)|x is in A and y is in B} where f maps A to B, but that's fairly tedious to work with so we use shortcuts like the above equation. The requirements to be a function are that it must map every point in the domain to a point in the range, and can only map each point in the domain to one point in the range. Using the formal definition above, that means for each a in A, there exists b in B such that (a,b) is in the set and for each a in A, there is only one b such that (a,b) is in the set.

An equation is simply something of the form X=Y where x and y are strings of characters that have to make sense, and the statement X=Y implies that X and Y evaluate to the same thing

 

[ScienceNerd36]

So an equation expresses the relationship between two things while a function allows you to deduce information about a domain?

 

[Дьявол]

Actually you can form equation of functions

Ex. f(a)-f(b)=0

 

[ScienceNerd36]

And that's displaying the relationship between two functions.

 

[Дьявол]

And that's displaying the relationship between two functions.

Yep, and every function is equation since,

''An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent)''

So f(x) is equal to x+5, or f(x)=x+5 or f(x)-x-5=0.

Regards.

 

[flatmaster]

Functions are a subset of all equations. A function is an equation with the restrictions mentioned earlier.

 

[CaffeineJunky]

An equation is simply a statement: "1 = 1" is an equation that is true. Equality itself is a relation on a set. In fact, equality is an example of an equivalence relation--a relation that satisfies certain properties.A function is a much more abstract mathematical object that acts as a map from one set to another by mapping each element in the domain set to an element in the codomain set. A function is a special case of a relation.

(x^2 + y^2 = 1 | x, y real numbers) is a relation on the R x R, but not a function.

 

[Elucidus]

An equation is a special type of relation called a comparison and is also a logical statement. It also needs a context (oftimes called a "domain" but not in the same sense as the domain of a function) which is usually some sort of Cartesian product of the sets from whence the unbound variables are selected.

The equation

$z^2=x^2+y^2-2xy\cos{\theta}$

is defined for all values x, y, z, and $\theta$ for which it makes sense.

The equation

$P(E) = P(E|F)\cdotP(F)+P(E|F^\prime)\cdotP(F^\prime)$

is defined for all events in a sample space.

A function is an ordered triple $\langle A, B, \Gamma_f\rangle$ where f is a function from the domain A to the codomain B and has graph $\Gamma_f$ that is the set $\{(a,f(a))|a\in A\}$. As mentioned before each argument a in A must have exactly one value b in B.

The major difference is that an equation is logical object, while a function is an ordered triple (a set theoretic object).

Many functions are defined via equations, but that is not necessary, and in fact it has been proven that equations cannot detail all functions (there are only a countable number of equations while there are uncountable functions).

--Elucidus

 

[Tac-Tics]

A function is a rule. It associates with each possible input a single output.

A function's definition often comes in the form of an equation, but it need not. For a polynomial, we use the equation definition: f(x) = x^2 - 1. For piecewise functions, we break the definition up over several equations. For example:

f(x) = sin(x) / x, when x /= 0
f(x) = 1, when x = 0

Other times, the function's rule might just be given orally. For example, the function which maps vectors to their length.

Equations do not always lead to a good function definition. The easiest example is $$x = y^2$$. Because there are two possible y's for each x (a positive one and a negative one). Luckily in this case, we have a notation we use for a very closely related function: $$f(x) = \sqrt{x}$$.

 

[ScienceNerd36]

So taking analytical geometry as an example:

What would be the key difference in the result between using a function to graph a line, such as "f(x)=3x-2", and using the equation of that line, which would be "3x-y-2"?

 

[MattC72]

3x-y-2 doesn't mean anything (In a geometry sense) unless you have an equivalence relation (In this case, an equals sign). So if you had

$$3x-y-2 = 0$$
then naturally you would have
$$y=3x-2$$
and, therefore, the same as before.

But, if you mean plotting
$$f(x,y) = z = 3x-y-2$$
then you will get a 3-D graph (with axis (x,y,z)).

However, back to your original question; I think everyone's answers are a little too complicated. And, from the way you worded your question, it doesn't sound like you need a textbook answer; it sound like you just need a brief description so that you can get your head around it. So here goes:

A function is an instruction to do something to an object. I think of functions as machines. So input is a load of fruit, output is a smoothie. You put something in, you get something (usually different) out.

Equations are a type of function. Put number(s) in, get number(s) out.

So,
$$Set \ of \ Functions \supset Set \ of \ Equations$$


Matt

p.s. If this post makes you want to drink smoothies, I take no responsibility for the havoc it'll cause your insides.

 

[Tac-Tics]

So taking analytical geometry as an example:

What would be the key difference in the result between using a function to graph a line, such as "f(x)=3x-2", and using the equation of that line, which would be "3x-y-2"?

I'm assuming you mean 3x - y - 2 = 0. By itself, 3x - y - 2 is just an expression. You need a verb (usually an equals sign) to make it an equation.

If you graph the function f in the standard way, you do get a line in the plane which is uniquely defined by the equation 3x - y - 2 = 0. The two are very intimately related.

But the graph isn't the function. And an equation isn't a function. And an equation isn't a graph. These are all distinct concepts.

The equation is a statement with two free variables.

A graph is a set of points. In this case, the set of points is given by the set { (x, y) for all x, y where 3x - y - 2 = 0 }.

The function, f, is the rule. Keep in mind that f and y are two completely, totally different things! In the equation, y is a real number, and f is a function! However, when you apply an argument to f, you get a real number out. And so, f(x) is a real number.

In this case (and many others in algebra and calculus), the three are practically identical. But there are examples of each that don't meet the requirements of the other:

A graph of a square represents no function.
A specially defined function such as f(x) = 1, if x is rational and 0 otherwise has no algebraic equation behind it.
You can also have functions between things that aren't the real numbers. For example, the derivative operator. The derivative is a function which acts on real functions. It has no graph. It has no algebraic equation behind it.

 

[Elucidus]

So,
$$Set \ of \ Functions \supset Set \ of \ Equations$$


Matt

Unfotunately the relationship between them is the reverse.

$x^2+y^2=1$ is an equation but does not describe either variable as a function of the other.

--Elucidus

 

[Tac-Tics]

3x-y-2 doesn't mean anything (In a geometry sense) unless you have an equivalence relation (In this case, an equals sign).

Also, equivalence relation means something different than what you've used the word for here.

An equivalence relation is any relation (or you can think of it as a binary predicate, LIKE an equals sign) which exhibits the most important properties of equality: reflexivity, symmetry, and transitivity.

 

[ScienceNerd36]

However, back to your original question; I think everyone's answers are a little too complicated. And, from the way you worded your question, it doesn't sound like you need a textbook answer; it sound like you just need a brief description so that you can get your head around it. So here goes:

A function is an instruction to do something to an object. I think of functions as machines. So input is a load of fruit, output is a smoothie. You put something in, you get something (usually different) out.

Equations are a type of function. Put number(s) in, get number(s) out.

Thank You! I understand now. And I'm going to make a smoothie.

 

[MattC72]

Unfotunately the relationship between them is the reverse.

$x^2+y^2=1$ is an equation but does not describe either variable as a function of the other.

--Elucidus

Is this true?

You are right that $x^2+y^2=1$ is not a bijective function, but the 2 variables are functions of each other.

 

[Tac-Tics]

Equations are a type of function. Put number(s) in, get number(s) out.

Thank You! I understand now. And I'm going to make a smoothie.

No smoothie, because this is plain false. Some equations lend themselves to functions. Some don't. People keep throwing $$x^2 = y^2$$ as a simple example.

 

[MattC72]

Also, equivalence relation means something different than what you've used the word for here.

An equivalence relation is any relation (or you can think of it as a binary predicate, LIKE an equals sign) which exhibits the most important properties of equality: reflexivity, symmetry, and transitivity.

Yes, you are right in your definition of an equivalence relation, as a special type of binary operation. But commonplace notation holds that equivalence relations are characterized by their individual "equals signs". For example;

$$\cong \ and$$~

 

[MattC72]

No smoothie, because this is plain false. Some equations lend themselves to functions. Some don't. People keep throwing $$x^2 = y^2$$ as a simple example.

In this example:
$$x^2 = y^2$$
is this not a function:
$$f: A \to A$$
defined by
$$f:x\to \pm x$$
?

 

[MattC72]

Just so you know, I'm not trying to be awkward or argumentative. I'm genuinely curious.

 

[Tac-Tics]

is this not a function:
$$f:x\to \pm x$$?

It is indeed not. Functions, by their definition, are single-valued. You put one thing in, you get one thing out. The closest you can get with $$x^2 = y^2$$ is to use TWO functions: f(x) = x and g(x) = -x. If you graph BOTH f and g, the result is the graph of this equation.

(Now, you might hear about multi-valued functions from time to time, but this is a bastardization, reflecting older terminology before the idea of a function was fully developed.)

Just so you know, I'm not trying to be awkward or argumentative. I'm genuinely curious.

You're asking the right questions. If you want to do anything beyond high school calculus, it's very helpful to understand these kinds of things. Some of it boils down to terminology, but the stuff that doesn't, you should always be skeptical about until you see it for yourself ;-)

 

[MattC72]

It is indeed not. Functions, by their definition, are single-valued. You put one thing in, you get one thing out. The closest you can get with $$x^2 = y^2$$ is to use TWO functions: f(x) = x and g(x) = -x. If you graph BOTH f and g, the result is the graph of this equation.

(Now, you might hear about multi-valued functions from time to time, but this is a bastardization, reflecting older terminology before the idea of a function was fully developed.)



You're asking the right questions. If you want to do anything beyond high school calculus, it's very helpful to understand these kinds of things. Some of it boils down to terminology, but the stuff that doesn't, you should always be skeptical about until you see it for yourself ;-)

Goodness me, that's quite an assumption. I've just graduated with a master's degree in mathematics Thank you very much. You should only assume your axioms! ;)

Now, I admit I did little more than group theory for the last 2 years but talking multivalued functions seem to be pretty standard practice in the current mathematical world.

 

[Tac-Tics]

You should only assume your axioms! ;)

My apologies. We do see questions like this coming from high school graduates pretty often. They come across the square root function. Their teachers lie to them and tell them the square root of x^2 is this stupid thing called "plus or minus ecks". They get all confused and start losing credit on assignments.

All they need is a healthy dose of set theory.

talking multivalued functions seem to be pretty standard practice in the current mathematical world.

Multi-valued functions are convenient for some handwaving arguments. When you can call multi-valued things functions, every functions becomes "invertible" which is nice. The only catch is that thinking in those kinds of terms, you can mess up when reasoning about unfamiliar problems. You end up finding a solution instead of all solutions.

(But when you're working with physics or applied mathematics, a solution is sometimes good enough).

 

[HallsofIvy]

My apologies. We do see questions like this coming from high school graduates pretty often. They come across the square root function. Their teachers lie to them and tell them the square root of x^2 is this stupid thing called "plus or minus ecks".[/itex]

I have never heard a teach say such a thing, even when I was myself in elementary or high school. I have many times heard teachers say that the solutions to the equation $x^2= a$ are $x= \pm \sqrt{a}$ which is the exact opposite of the "lie" you say they tell. If $\sqrt{a}$ were both positive and negative you would not need the "$\pm$.

[/itex] They get all confused and start losing credit on assignments.

All they need is a healthy dose of set theory.



Multi-valued functions are convenient for some handwaving arguments. When you can call multi-valued things functions, every functions becomes "invertible" which is nice. The only catch is that thinking in those kinds of terms, you can mess up when reasoning about unfamiliar problems. You end up finding a solution instead of all solutions.

(But when you're working with physics or applied mathematics, a solution is sometimes good enough).

 

[Tac-Tics]

I have never heard a teach say such a thing, even when I was myself in elementary or high school.

Hah. They don't really "lie". But you know that it's one of the most common questions here.

 

[Elucidus]

Hmmm. I seem to have made a hash of things by stating two contradictory statements - that there are more functions than equations and that functions are a subset of equations (these cannot both be true). I reflected on the original question since I left work and felt the need to clarify my original comments and to also re-adress the issue.

My first remark was intended to comment on the idea of a "rule of assignment." Many textbooks define a function like "A function from a domain A to a codomain B is a rule that assigns a single value from B to each argument from A," or some such. The problem is with the whole "rule" concept. It has been shown that there are only a countable number of rules (since they must be determinate). Unfortunately it is easy to construct a function space that is demonstrably uncountable. This means that there are functions that cannot be represented by rules.

Consider for simplicity real-valued functions of a real variable. The function can be defined on any subset of the real line and for any argument in the domain it can have its choice of real value. If $\mathfrak{c}$ is the cardinality of the reals then there are $\mathfrak{c}^\mathfrak{c}$ real-valued functions of a real variable (which is definitely uncountable).

A function's rule of assignment is often given in equation form (i.e. f(x) = x² + 1). But this is really shorthand for:

$$f=(x\mapsto x^2+1):\mathbb{R}\rightarrow\mathbb{R}$$

(Note: Even this notation cannot capture all function representations.)

The graph of a function (in the set theoretic sense, not the diagramatic sense) is a certain type of set of ordered pairs and is a subset of $A \times B$ where the function is from A to B.

Equations are also certain subsets of a Cartesian product. They are equivalence relations and are frequently not function-like (and vice-versa).

Multi-valued functions are technically functions from a set A to the power set of another set B. The image is still a single object, but that object has potentially more than one member.

The natural logarithm of a complex number is an example. Since $e^{i\theta}=e^{i(\theta+2\pi)}$ then the logarithm must have an infinite number of values. The image of any complex number would then be the set of all such values. So the complex logarithm is

$$log=(z\mapsto \{ln|z|+i(arg(z)+2n\pi):n\in \mathbb{N}\}):\mathbb{C}\rightarrow 2^{\mathbb{C}}$$

Statements like "let f(x) = 2x - 3..." is an abuse of notation since it omits the domain, codomain and isn't actually a mapping. This abuse has been perpetrated for so long that it is tradition and getting people to stop is counterproductive.

Getting back to the original question:

This is an example of an equation:

sin(x)cos(x) + cos²(x)=0

This is a function:

$f=(x\mapstox^3-x):\mathbb{R}\rightarrow\mathbb{R}$

One is a logical comparison of two things and is either true or false for a particular list of values for its variables, while the other is a mapping that takes an input and produces an output. The function may be represented or defined using an equation or it might not.

Hopefully I haven't muddled the issue farther.

--Elucidus

 

[Tac-Tics]

Unfortunately it is easy to construct a function space that is demonstrably uncountable. This means that there are functions that cannot be represented by rules.

I think this is way beyond the scope of the OP's question. Also, using the standard set theoretical definition of a function, there's nothing wrong with rules of infinite complexity. (Think piece-wise functions with an infinite number of cases). It's no different from allowing infinite decimal representations for real number. At the very least, you have to accept that for every real number c, there is a constant function f(x) = c, so there are at least as many functions (cardinalitywise) as there are reals.

 

[ScienceNerd36]

I've never felt so belittled by mathematics. I think I'll just stick to physics. See you on another thread.

And thanks for your help:)