By D. E. Rydeheard
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They provide an ‘interface’ to the representation. With this separation, we may change the representation and redefine the operations, without needing to change programs using the type. To make this effective, the representation should be inaccessible outside its definition. The operations on a type determine how much of the representation is available and so, where there is superfluous structure in a representation, this may be hidden. This separation of representation from use is called ‘data abstraction’ and is a fundamental idea in the organization of programs.
Tn ∈ TΩ (X) We define composition using application (gf )(x) = g(f (x)). Also, for each set X the identity substitution is defined by i X (x) = x (unless X is empty in which case the identity is the empty function). Define TΩ to be the category whose objects are sets and whose arrows are substitutions. This is indeed a category under the composition and identities above. The full subcategory of finite sets is denoted T Ω F in . This category provides a basis for a categorical treatment of equational deduction.
To print sample results, you will need a function converting sets to strings of characters for display. Moreover, to define powersets you need equality on finite sets. Exercise∗ 10. Sorting This is an exercise in programming an algorithm for sorting a list of items, which support a total order, into a non-descending sequence. The algorithm is called ‘tree sort’ and works by inserting items successively into an ‘ordered’ tree and then flattening the resultant tree. Those unfamiliar with the algorithm should consult a reference such as [Knuth 73].