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?
2026-05-14 10:07:44.1778753264
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What is the intersection between the set of all expressions, of all equations and of all functions?
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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
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.