Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 261 — #269
8.3. The Logic of Sets 261
We’re not going to be studying the axioms of ZFC in this text, but we thought
you might like to see them—and while you’re at it, get some more practice reading
and writing quantified formulas:
Extensionality. Two sets are equal iff they are members of the same sets:
x D y IFF .8z: z 2 x IFF z 2 y/:
Pairing. For any two sets x and y, there is a set fx; yg with x and y as its only
elements:
8x; y9u8z: Œz 2 u IFF .z D x OR z D y/
Union. The union u of a collection z of sets is also a set:
8z9u8x: .x 2 u/ IFF .9y: x 2 y AND y 2 z/
Infinity. There is an infinite set. Specifically, there is a nonempty set x such that
for any set y 2 x, the set fyg is also a member of x.
Subset. Given any set x and any definable property of sets, there is a set y containing
precisely those elements in x that have the property.
8x9y8z: z 2 y IFF Œz 2 x AND .z/
where .z/ is a formula of set theory. 6
Power Set. All the subsets of a set form another set:
8x9p8u: u x IFF u 2 p:
Replacement. Suppose a formula of set theory defines the graph of a total function
on a set s, that is,
8x 2 s 9y: .x; y/;
and
8x 2 s 8y; z: Œ.x; y/ AND .x; z/ IMPLIES y D z:
Then the image of s under that function is also a set t. Namely,
9t8y: y 2 t IFF Œ9x 2 s: .x; y/:
6 This axiom is more commonly called the Comprehension Axiom.