# What is the intersection between the set of all expressions, of all equations and of all functions?

I am studying the definition of mathematical expression, of equation and of function and I want to draw a venn diagram with the intersection between the set of these objects. Some people say every function is an equation, although not every equation is a function, so would it be precise if I draw the set of functions inside the set of equations and the set of equations inside the set of expressions?

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Most functions don't have equations. Many equations are only implicit equations. Expressions have nothing to do with this. – Karolis Juodelė Nov 6 '13 at 21:11
@KarolisJuodelė In the Penguin Dictionary of Mathematics it is written that expression is any mathematical form expressed symbolically, as in an equation, polynomial, etc. So if all equations are expressions why wouldn't have something to do with this? – m.Os Nov 6 '13 at 21:26
I see. Wikipedia says that equation is two expressions and a = sign. So I assumed. – Karolis Juodelė Nov 6 '13 at 22:25

Expression is a thing written down. Relation is a set of tuples. "$2$" is not a relation, all equations represent the set of their solutions, most relations don't have expressions (there are much more relations than expressions).

• Equation $=$ Expression $\cap$ Relation

A function is a special kind of relation where a subset of variables uniquely identify the remaining ones. "$x = y$" is a function, but as with relations, not all of them have expressions. "$x^2 + y^2 = 1$" is an equation but not a function - it may be called an implicit function which is the same as equation.

• Function $\subsetneq$ Relation
• Function $\cap$ Equation $\subsetneq$ Function
• Function $\cap$ Equation $\subsetneq$ Equation
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In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.

Thus it would be an equation if we picked out a specific $x$ in $X$, constructed a bijective function $f: X \to Y$ where $Y$ is a set and it contains $f(x)$ and there exists an inverse function $f^{-1}$ such that it maps $f(x) \to x$.

An expression is a manner of conducting ideas on paper. Thus, if your ideas made sense to the other person, and it consisted of bijective functions, then we can say that your expression has a bijective function. I think this is what you are looking for.

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