# Is a binary function a set of ordered pairs?

The definition of a function, according to the book I have, is: a function is a set of ordered pairs, no two of which have the same first component. Thus an ordered pair in this set can be expressed as $$\pmb{(a,b)}$$ or $$\pmb{(a, f(a))}$$.

However, a binary function, from what I've read, implies a ternary relation, expressed as $$\pmb{(a, b, f(a,b))}$$, which is not an ordered pair but a triplet. Thus does not satisfy the definition of a function? Shouldn't it be expressed as ($$\pmb{(a,b)}$$, $$\pmb{f(a,b)}$$)? Because only then it will become an ordered pair. Makes sense because of the definition. If we defined the relation of sets $$\pmb{A}$$, $$\pmb{B}$$ and, let's say, $$\pmb{C}$$ as the cartesian product, then we get $$\pmb{A \times B \times C}$$ which implies taking the product of a set of ordered pairs, $$\pmb{A \times B}$$, and a set, $$\pmb{C}$$, whose elements aren't ordered pairs, so the possible combinations will be of the form ($$\pmb{(a,b)}$$, $$\pmb{f(a,b)}$$) or ($$\pmb{(a,b)}$$, $$\pmb{c}$$) where $$\pmb{a}$$, $$\pmb{b}$$ and $$\pmb{c}$$ are elements of $$\pmb{A}$$, $$\pmb{B}$$ and $$\pmb{C}$$ respectively, satisfying the definition previously stated and not ($$\pmb{a}$$, $$\pmb{b}$$, $$\pmb{c}$$), or is it a mere notation meant to simplify ($$\pmb{(a,b)}$$, $$\pmb{f(a,b)}$$)?

To be properly formal a binary function $$f: A \times B \to C$$ is a collection of pairs $$(x,y)$$ with $$x \in A \times B$$ and $$y \in C$$ such that each $$x$$ is the first element of exactly one pair. We write $$f(x)$$ for the second element of the pair.
Since elements of $$A \times B$$ have the form $$(a,b)$$ the function is a collection of pairs $$((a,b),y)$$. Then to be really REALLY formal you should write the outputs as $$f((a,b))$$ rather than $$f(a,b)$$.
• Notation $f(a,b)$ for the output since there is no ambiguity. For pairs write $((a,b),f(a,b))$. although I cannot imagine many situations you will need to write down pairs in the future. Feb 18 at 23:45