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text: 'A **partial function** is like a [-3jy] $f: A \\to B$, but where we relax the requirement that $f(a)$ must be defined for all $a \\in A$.\nThat is, it must still be the case that "$a = b$ and $f(a)$ is defined" implies "$f(b)$ is defined and $f(a) = f(b)$", but now we no longer need $f(a)$ to be defined everywhere.\nWe can write $f: A \\rightharpoonup B$ %%note:In LaTeX, this symbol is given by `\\rightharpoonup`.%%to denote that $f$ is a partial function with **domain** $A$ and **codomain** $B$: that is, whenever $f(x)$ is defined then we have $x \\in A$ and $f(x) \\in B$.\n\nThis idea is essentially the "flip side" to the distinction between the [-3lv] dichotomy.\n\n# Implementation in set theory\n\nJust as a function can be implemented as a set $f$ of ordered pairs $(a, b)$ such that: \n\n- every $x \\in A$ appears as the first element of some ordered pair in $f$\n- if $(a, b)$ is an ordered pair in $f$ then $a \\in A$ and $b \\in B$\n- if $(a, b)$ and $(a, c)$ are ordered pairs in $f$, then $b=c$\n\nso we can define a *partial* function as a set $f$ of ordered pairs $(a,b)$ such that:\n\n- if $(a, b)$ is an ordered pair in $f$ then $a \\in A$ and $b \\in B$\n- if $(a, b)$ and $(a, c)$ are ordered pairs in $f$, then $b=c$\n\n(That is, we omit the first listed requirement from the definition of a *bona fide* function.)\n\n# Relationship to Turing machines\n[todo: maybe this should be under Turing machine rather than partial function?]\n\nMorally speaking, every [5pd Turing machine] $\\mathcal{T}$ may be viewed as computing some function $f: \\mathbb{N} \\to \\mathbb{N}$, by defining $f(n)$ to be the state of the tape after $\\mathcal{T}$ has been allowed to execute on the tape which has been initialised with the value $n$.\n\nHowever, if $\\mathcal{T}$ does not terminate on input $n$ (for example, it may be the machine "if $n = 3$ then return $1$; otherwise loop indefinitely"), then this "morally correct" state of affairs is not accurate: how should we define $f(4)$?\nThe answer is that we should instead view $f$ as a *partial* function which is just undefined if $\\mathcal{T}$ fails to halt on the input in question.\nSo with the example $\\mathcal{T}$ above, $f$ is the partial function which is only defined at $3$, and $f(3) = 1$.',
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