k This axiom is sometimes stated only for setsxconsisting of pairwise disjoint elements, and then proved for all sets.. 5.2.2 Basic operations on sets and their properties I hereafter de
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This axiom is sometimes stated only for setsxconsisting of pairwise disjoint elements, and then proved for all sets
The axiomatic system ZF is a very rich theory within which most of the contemporary mathematics can be built ZF with the added axiom AC is denoted ZFC While AC looks very natural and almost obvious, it turns out to have some quite unintuitive, even paradox-ical, consequences, so it has always been treated with special care For more on ZF and ZFC, see the references I do not delve into the foundations of set theory here, but only use that theory to illustrate logical reasoning and mathematical proofs
5.2.2 Basic operations on sets and their properties
I hereafter denote some sets with capital lettersA, B, C, , X, Y, Zand elements of those sets with lower-case lettersa, b, c, , x, y, z This should hopefully be intuitive and help to avoid confusion
The existence of the empty set ∅ and some of the basic operations on sets, namely
union∪ and powerset of a set P, are postulated with axioms of ZF The other basic
oper-ations – intersection∩, difference −, and Cartesian product × – can also be defined and
justified within ZF
I list a number of properties of these operations, to be formalized in first-order logic and proved as exercises First, we make some important general remarks on defining objects
in ZF In order to formally define an operation or relation on sets, we can write its defining equivalence inL ZF as we did for⊆ above For instance, we can define the proper subset
relation⊂ as
∀ x ∀ y(x ⊂ y ↔ x ⊆ y ∧ x = y)
and the intersection∩ by:
∀ x ∀ y ∀ z(z ∈ ( x ∩ y)↔ ( z ∈ x ∧ z ∈ y))
As an exercise, use the separation axiom SEP to prove the existence ofx ∩ y For that, write its definition in a set-builder notation and note that it can now be relativized to already “existing” sets
Likewise, the Cartesian product of two sets A, B can formally be defined in set-builder notation as
A × B:={( a, b)| a ∈ A, b ∈ B }
where (a, b) is the ordered pair consisting of aas a first element and b as a second element A strange-looking but formally correct (see exercise) and commonly accepted way to define an ordered pair as a set is
(a, b) :={{ a } , { a, b }}.
As an exercise, using axioms of ZF show that the Cartesian product of any two sets exists
In the next four propositions, note the close parallel between the properties of the logical connectives⊥ , ∧ , ∨, and −, where A − B :=A ∧ ¬ B, and of their set-theoretic coun-terparts∅ , ∩ , ∪, and −.