What are the different types of equations and their names?

[pairofstrings]

TL;DR Summary

There can be a expression on LHS and RHS, 1 on LHS expression on RHS, y on LHS expression on RHS.

Hi.
What is the name of the equation whose LHS and RHS are expressions?
What is the name of the equation whose LHS is constant like '1' and RHS is an expression?
What is the name of the equation whose LHS is a variable like 'y' and RHS is an expression?

What is the name of the equations in the below statements?
1 = expression --?
y = expression --?
expression = expression --?

Are there any other variants that I am missing?
Thanks.

 

[BvU]

Hi,

This is a physics forum. They are all equations. No subspecies like in biology.
Why should they have names ?

$\$

 

[pairofstrings]

Why should they have names ?

I thought they might have names like several other different types of equations like Logarithmic, Simultaneous, Cubic, Quadratic, Trinomial, Binomial, Monomial, Polynomial, Trigonometric, Radical, Exponential, Rational, Linear, Cartesian, Polar, Algebraic, Nonparametric, Parametric. These are equations. So, if there so many different types of equation I thought those three equations could have three more names. :)

 

[fresh_42]

Summary:: There can be a expression on LHS and RHS, 1 on LHS expression on RHS, y on LHS expression on RHS.

Hi.
What is the name of the equation whose LHS and RHS are expressions?
What is the name of the equation whose LHS is constant like '1' and RHS is an expression?
What is the name of the equation whose LHS is a variable like 'y' and RHS is an expression?

What is the name of the equations in the below statements?
1 = expression --?
y = expression --?
expression = expression --?

Are there any other variants that I am missing?
Thanks.

Names are given if it a) helps to summarize complex conditions, b) an object occurs repeatedly, or c) contributes to clarification. They are normally context-sensitive. E.g., the word ring can have various meanings. Sure, we have named formulas (Pythagoras, Euler, etc.) but these refer to specific equations, not to some sort of equations. $1=f(x,y)$ is a contour line, a level equation. However, note the context-sensitivity: If we are not wandering through calculus, but e.g. through linear algebra, then $1=f(x,y)$ might simply refer to some invertible matrices, a determinant, or whatever.

Hence context is the primary condition to be specified before we talk about names.

 

[pairofstrings]

Hence context is the primary condition to be specified before we talk about names.

The context is that I want to study a situation where there is equation like: expression = expression.

This "expression = expression" may not have a name..

 

[BvU]

What changes for you if it has a name ? Do you want to look it up with google ?

 

[Ibix]

The context is that I want to study a situation where there is equation like: expression = expression.

$$\begin{eqnarray*}
f(x)&=&g(x)\\
\frac{f(x)}{g(x)}&=&1
\end{eqnarray*}$$So isn't it a matter of choice whether you have a pure number on one side or not?

 

[fresh_42]

The context is that I want to study a situation where there is equation like: expression = expression.

Please define "expression", rigorously! What is your alphabet, what is your syntax, and what is your grammar?

 

[hutchphd]

I believe the OP is searching for the term "implicit function" or "implicitly defined function".
Are we all on nasty pills today?

 

[fresh_42]

I believe the OP is searching for the term "implicit function" or "implicitly defined function".
Are we all on nasty pills today?

I don't think so. Definitions of used technical terms are the basis for any scientific discussion. Implicit functions are a guess as any other guess before. Expression isn't well-defined, so it's legitimate to ask what is meant by it.

 

[hutchphd]

Maybe if he knew the definition he wouldn't be asking the question??

 

[pairofstrings]

What changes for you if it has a name ?

Nothing much and I will move on little sad.

Do you want to look it up with google ?

I say that there could be three distinct artifacts:

• expression = 1
• expression = y
• expression = expression

The first is a object confined within 1 unit on a graph.
The second one is a function.
The third -- ?

This the third is what I am after.

Please define "expression", rigorously! What is your alphabet, what is your syntax, and what is your grammar?

I say that "expression" is some finite number of terms that can exist on either side of equals-to and these terms can be numbers, variables, expression(?).

Thanks.

 

[fresh_42]

Let us assume you mean functions on real numbers: $f(x)=y\, : \,\mathbb{R} \longrightarrow \mathbb{R}$.

Then $f(x)=1$ can be viewed as $f(x)-1=0$. If $x\in \mathbb{Z}$ and $f(x)$ are linear functions, then the system is called Diophantic.

If we consider the possible values $x$ then we call them - the $x$ - zeros, or roots; algebraic variety under other circumstances ($f$ polynomial, possibly other fields than $\mathbb{R}) .$

Depending on the structure of $f(x):=g(x)$ as a definition of $f$ by $g$, it is called simply a function, polynomial, exponential, constant, continuous, $n$-times differentiable, or whatever.

I do not see the difference between expression = y and expression=expression.

It all depends on context and aim.

 

[BvU]

Nothing much and I will move on little sad.

no need !

I say that there could be three distinct artifacts:

• expression = 1
• expression = y
• expression = expression

The first is a object confined within 1 unit on a graph.
The second one is a function.
The third -- ?

This the third is what I am after.

Unfortunately I see no difference at all between the three

• expression - 1 = 0 $\qquad\qquad\Leftrightarrow\qquad$ expression = 0
• expression - y = 0 $\qquad\Leftrightarrow\qquad$ expression = 0
• expression - expression = 0 $\qquad\Leftrightarrow\qquad$ expression = 0

 

[fresh_42]

There is a name for expression with a colon.
$f(x):=g(x)$ ($f$ is defined by $g$) or $f(x)=:g(x)$ ($g$ is defined by $f$) are definitions.

 

[Mark44]

Unfortunately I see no difference at all between the three

• expression - 1 = 0 ⇔ expression = 0
• expression - y = 0 ⇔ expression = 0
• expression - expression = 0 ⇔ expression = 0

I agree completely. The original classifications -- expression = 1, expression = y, expression = expression -- are not useful in categorizing equations.

 

[DaveE]

 

[pairofstrings]

Let us assume you mean functions on real numbers: f(x)=y:R⟶R.

Here the equation 'y' in terms of 'x' is called a function..
So, shouldn't there be a name if the equation is in both x, y?

 

[fresh_42]

Here the equation 'y' in terms of 'x' is called a function..
So, shouldn't there be a name if the equation is in both x, y?

$y(x)$ in $f(x,y(x))=\ldots$ is called an implicit function. $f(x,y)=\ldots$ is still a function.

To make it worse: even function is already an implicit assumption. It could as well be just a relation.

 

[pairofstrings]

y(x) in f(x,y(x))=…

The above equation 'y' has terms in 'x' and it is called function?
Equation even with x, y variables on either sides of the equation is a function?

 

[fresh_42]

The above equation 'y' has terms in 'x' and it is called function?
Equation even with x, y variables on either sides of the equation is a function?

Yes and yes. There is no rule on how many variables a function has. O.k., it's usually a finite number, but even infinitely many variables are sometimes regarded.

If $y$ depends on $x$, in notation: $y=y(x)$, and $y$ also occurs as a variable of a function, then $f(y)=f(y(x))=f(x,y(x))$ the function automatically depends on $x$, too. If only $f$ is given, then we say that $y(x)$ is implicitly given, namely with the help of $f$.

 

[pairofstrings]

But there is something called "Vertical Line Test" that says if the graphs has more than one intersection then it is said that the equation is not a function.
What is your saying on this?

Thanks.

 

[fresh_42]

But there is something called "Vertical Line Test" that says if the graphs has more than one intersection then it is said that the equation is not a function.
What is your saying on this?

Thanks.

Yes. In that case we have appointed to $y$-values to the same $x$-value, i.e. $y_1=f(x)=y_2$. The vertical line at $x$ has two intersections with $f$, namely at the two heights $y_1$ and $y_2$.

But even my notation as $y_1=f(x)=y_2$ is already wrong. You cannot write it as a function. If it appoints two values, then it is a relation. Every function is a relation, but not every relation is a function. A relation is a subset of $X\times Y,$ the pairs of all $x-$values and all $y-$values. In this case we would call the subset $R$ for relation and write $(x,y_1)\in R$ and $(x,y_2)\in R.$

 

[pairofstrings]

Is equation of a circle: $x^{2}+y^{2}=1$ also a function?

 

[fresh_42]

It depends on how you define it. It is the contour line of the function $f(x,y)=x^2+y^2$ at level $1.$ In this case our function goes $f\, : \,\mathbb{R}^2\longrightarrow \mathbb{R}\, , \,(x,y)\mapsto (x^2+y^2).$

However, it is no function of the reals to the reals. The vertical line test fails. We could talk about implicit functions in this case, but I'm afraid that it might confuse you. For short: the circle is locally a function if $x\neq \pm 1$. In a small neighborhood around a point, we have a function since e.g. the lower half of the circle isn't in that neighborhood. If $x \in (0,1)$ and $y>0$, then $y=\sqrt{1-x^2}$. Similar if $y<0$, then we get $y=-\sqrt{1-x^2}$. These are local functions. The entire circle as subset of $\mathbb{R}\times \mathbb{R}$ is a relation, no function.

 

[eltodesukane]

expression = expression
is the same as
0 = expression - expression