Popular when learning Set Theory

A function $f:A\to B$ is a subset $f\subset A\times B$ where 

• Domain $\forall x\in A,\exists y\in B, (x,y)\in f$
• Well-defined $\forall (x,y),(x,z)\in f,(y=z)$

So $(x,y)\in f$ means $y$ is unique to $x$ can we call it the output denoted $$f(x)=y.$$

Can't start with functions as sets

• What is $A\times B$?
• Is it $\{(a,b)\mid a\in A, b\in B\}$? So what is $(a,b)$?
• $(a,b)$ is gadget you can get both $a$ and $b$ from, i.e. there is a function $(a,b)\mapsto a$...
• Circular to define functions as subsets of $A\times B$
• Axiom of Specification: all $\{x\mid S(x)\}$ are sets.  Set Theory axioms assume functions already exist.
• Cardinality, functors, and more are functions but by Russell's paradox their domains are not sets.

Replace variables

$f(x)=x+2$

If $P$ and $P\Rightarrow Q$ then $Q$

In $\lambda$-notation

$\lambda x.2x$

Compose atomic functions

CMP AX BX
DIV CX DX
JMP RX

println(toString(541));

In combinator notation

$S(BBS)(KK)$

Replace variables

Church (1930) $\lambda$-calculus

• Laws of substitution

Compose atomic functions

Schoenfinkel (1920)

Curry (1927) Combinators

• Three primitive functions, Identity $I$, Constant $K$, and Strong composition $S$