Mathematics for Computer Science
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“mcs” — 2017/3/3 — 11:21 — page 280 — #288<br />
280<br />
Chapter 8<br />
Infinite Sets<br />
(c) The Foundation axiom of set theory says that 2 is a well founded relation<br />
on sets. Express the Foundation axiom as a <strong>for</strong>mula of set theory. You may use<br />
“member-minimal” and “isempty” in your <strong>for</strong>mula as abbreviations <strong>for</strong> the <strong>for</strong>mulas<br />
defined above.<br />
(d) Explain why the Foundation axiom implies that no set is a member of itself.<br />
Homework Problems<br />
Problem 8.32.<br />
In writing <strong>for</strong>mulas, it is OK to use abbreviations introduced earlier (so it is now<br />
legal to use “D” because we just defined it).<br />
(a) Explain how to write a <strong>for</strong>mula, Subset n .x; y 1 ; y 2 ; : : : ; y n /, of set theory 9<br />
that means x fy 1 ; y 2 ; : : : ; y n g.<br />
(b) Now use the <strong>for</strong>mula Subset n to write a <strong>for</strong>mula, Atmost n .x/, of set theory<br />
that means that x has at most n elements.<br />
(c) Explain how to write a <strong>for</strong>mula Exactly n of set theory that means that x has<br />
exactly n elements. Your <strong>for</strong>mula should only be about twice the length of the<br />
<strong>for</strong>mula Atmost n .<br />
(d) The direct way to write a <strong>for</strong>mula D n .y 1 ; : : : ; y n / of set theory that means<br />
that y 1 ; : : : ; y n are distinct elements is to write an AND of sub<strong>for</strong>mulas “y i ¤ y j ”<br />
<strong>for</strong> 1 i < j n. Since there are n.n 1/=2 such sub<strong>for</strong>mulas, this approach<br />
leads to a <strong>for</strong>mula D n whose length grows proportional to n 2 . Describe how to<br />
write such a <strong>for</strong>mula D n .y 1 ; : : : ; y n / whose length only grows proportional to n.<br />
Hint: Use Subset n and Exactly n .<br />
Exam Problems<br />
Problem 8.33. (a) Explain how to write a <strong>for</strong>mula Members.p; a; b/ of set theory<br />
10 that means p D fa; bg.<br />
Hint: Say that everything in p is either a or b. It’s OK to use sub<strong>for</strong>mulas of the<br />
<strong>for</strong>m “x D y,” since we can regard “x D y” as an abbreviation <strong>for</strong> a genuine set<br />
theory <strong>for</strong>mula.<br />
A pair .a; b/ is simply a sequence of length two whose first item is a and whose<br />
second is b. Sequences are a basic mathematical data type we take <strong>for</strong> granted, but<br />
when we’re trying to show how all of mathematics can be reduced to set theory, we<br />
9 See Section 8.3.2.<br />
10 See Section 8.3.2.
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