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I understand that the definition of a function, mathematically, goes beyond that of explicit functions that most physicists and engineers deal with. However, I came across the definition for a function in a specific book which seems circular. It goes as follows.

First the book states in English that essentially a function is an assignment from one set to another. Then they define the graph (Gr) of a function to be Gr( f ) = {(x, f (x)): x ∈ X}.

After introducing Gr, then the book states, "we may now give an entirely rigorous definition of a function by saying that a function is a subset, G, of X × Y which satisfies the condition that for each element x of X there exists exactly one y in Y such that (x, y) is in G."

But, the y in (x,y) is defined as f(x)....so is this not circular?

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    $\begingroup$ No, the $y$ in $(x,y)$ isn't defined at all, it's just $y$. Then, we define $f(x)$ to be this $y$. $\endgroup$ – Gerry Myerson Nov 20 '17 at 1:50
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I'll try to elaborate on what the book is saying. There are many potential ways to define the notion of a function $f : X \to Y$. Whichever definition you give, it is possible to define the set $$\mathrm{Gr}(f) = \{ (x,f(x)) \mid x \in X \} \subseteq X \times Y$$ No matter what definition of 'function' you take, this set has the property that, for all $x \in X$, there is a unique $y \in Y$ such that $(x,y) \in \mathrm{Gr}(f)$; in particular, for a given value of $x \in X$, this unique value of $y$ is precisely $f(x)$.

Suppose now that $G \subseteq X \times Y$ is an arbitrary subset such that, for all $x \in X$, there is a unique $y \in Y$ such that $(x,y) \in G$. Then $G=\mathrm{Gr}(f)$ for a unique function $f : X \to Y$, since $f$ is determined by letting $f(x)$ be equal to the unique element of $y$ for which $(x,y) \in G$.

Thus there is a bijective correspondence between

  • Functions $f : X \to Y$; and
  • Subsets $G \subseteq X \times Y$ such that, for all $x \in X$, there is a unique $y \in Y$ with $(x,y) \in G$.

What the author then (implicitly) claims is that in light of this observation, we can simply say that a function $f : X \to Y$ is a subset of $f \subseteq X \times Y$ satisfying the above condition. Given $x \in X$, you can then recover the value $f(x)$ as being the second component of the only pair in the subset whose first component is $x$.

It's not a circular definition, but it does mean that $f=\mathrm{Gr}(f)$.

Side-note: I have reasons for disliking the convention of identifying functions with their graphs, but they're outside the scope of your question, so I'll refrain from ranting.

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