By D. E. Rydeheard

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G(f (x)), c) A subcategory of Set is that of finite sets, FinSet, whose arrows are again typed total functions. There are other categories whose objects are sets. For instance, we may consider arrows to be not total functions but partial functions to get a category SetP f . We may go further and consider objects again to be sets but arrows to be relations between sets (labelled with their source and target sets). A relation r : a → b is a subset of the cartesian product a × b. The composition sr of r : a → b with s : b → c is defined by sr = {(x, z) : ∃y ∈ b .

Corresponding to the deduction a ∧ b a, we have the rule πa,b : a ∧ b → a and likewise the rule πa,b : a ∧ b → b These are elimination rules for conjunction. The introduction rule, constructing arrows from arrows, is f : c → a, g : c → b f, g : c → a ∧ b To make this structure into a category, we assume the existence of identity entailments a a: ia : a → a and an associative composition of proofs, f : a → b, g : b → c gf : a → c for which the identity arrows are indeed identities. e. a ∧ b b ∧ a.

7 and define several operations on sets. These operations will arise in the following chapters, when we consider internal structure within the category of finite sets. Let us begin with a definition of a function which takes the image of a finite set through a function: fun image(f)(s) = if is_empty(s) then emptyset else let val (x,s’) = singleton_split(s) in union(singleton(f(x)),image(f)(s’)) end Now define the following operations: 1. The disjoint union of finite sets. 2. The cartesian product of two finite sets.