Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 281 — #289
8.4. Does All This Really Work? 281
need a way to represent the ordered pair .a; b/ as a set. One way that will work 11
is to represent .a; b/ as
pair.a; b/ WWD fa; fa; bgg:
(b) Explain how to write a formula Pair.p; a; b/, of set theory 12 that means p D
pair.a; b/.
Hint: Now it’s OK to use subformulas of the form “Members.p; a; b/.”
(c) Explain how to write a formula Second.p; b/, of set theory that means p is a
pair whose second item is b.
Problems for Section 8.4
Homework Problems
Problem 8.34.
In this problem, structural induction and the Foundation Axiom of set theory provide
simple proofs about some utterly infinite objects.
Definition. The class of ‘recursive-set-like” objects, Recs, is defined recursively as
follows:
Base case: The empty set ; is a Recs.
Constructor step: If P is a property of Recs’s that is not identically false, then
is a Recs.
fs 2 Recs j P.s/g
(a) Prove that Recs satisfies the Foundation Axiom: there is no infinite sequence
of Recs, r o ; r 1 ; : : : ; r n 1 ; r n ; : : : such that
Hint: Structural induction.
(b) Prove that every set is a Recs. 13
Hint: Use the Foundation axiom.
: : : r n 2 r n 1 2 : : : r 1 2 r 0 : (8.13)
11 Some similar ways that don’t work are described in problem 8.29.
12 See Section 8.3.2.
13 Remember that in the context of set theory, every mathematical datum is a set. For example,
we think of the nonnegative integers, 0,1,2,. . . , as a basic mathematical data type, but this data type
could be understood as just referring to the sets ;; f;g; ff;gg; : : : .