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In this situation, we say that f has functionality A → B_| and f(a) = −| . The use of the codomain B_| instead of B stands as a kind of warning: in the process of computing a Bvalue, nontermination could occur. Including −| as a value is an alternative to using a theory of partial functions. ) A function f that is undefined at argument a has the property f(a) = −| . In addition to dealing with undefinedness as a real value, we can also use −| to clearly state what happens when a function receives a nonterminating value as an argument.

Nonetheless, it is important to remember that an equation f(x) = α, for f : A → B, represents a function. The actual function is determined by a form of evaluation that uses substitution and simplification. To use f’s equational definition to map a specific a0 ∈ A to f(a0 )∈ B, first, substitute a0 for all occurrences of x in α. The substitution is represented as [ a0 /x]α. Second, simplify [ a0 /x]α to its underlying value. Here is the process in action: to determine the the value of add(2,3), we first substitute 2 for m and 3 for n in the expression on the right-hand side of add’s equation, giving add(2,3) = [3/n][2/m]m+n = 2+3.

C. d. e. make-singleton(add(3,2)) ∪ { 4 } add(snd(duplicate(4)), 1) which-part(inIN(add(2,0))) ([ 3|→ { 4 } ]make-singleton)(2) ([ 3|→ { 4 } ]make-singleton)(3) 13. For the equational definition fac(n) = (n=0) → 1 [] n∗fac(n−1), show that the following properties hold (hint: use mathematical induction): a. For all n ∈ IN, fac(n) has a unique value, that is, fac is a function. b. For all n ∈ IN, fac(n+2) > n. 14. List the elements in these domains: a. b. c. d. (IB × IB)_| IB_| × IB_| (IB × IB) + IB (IB + IB)_| Exercises 33 e.

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