General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

32 CHAPTER 3. ELEMENTARY DISCRETE MATH
We can represent sets by
listing the elements within curly brackets: e.g. {a, b, c}
describing the elements via a property: {x | x has property P }
stating element-hood (a ∈ S) or not (b ∈ S).
Axiom 57 Every set we can write down actually exists! (Hidden Assumption)
Warning: Learn to distinguish between objects and their representations!({a, b, c} and {b, a, a, c} are different representat
©: Michael Kohlhase 45
Indeed it is very difficult to define something as foundational as a set. We want sets to be collections
of objects, and we want to be as unconstrained as possible as to what their elements can be. But
what then to say about them? Cantor’s intuition is one attempt to do this, but of course this is
not how we want to define concepts in math.
So instead of defining sets, we will directly work with representations of sets. For that we only
have to agree on how we can write down sets. Note that with this practice, we introduce a hidden
assumption: called set comprehension, i.e. that every set we can write down actually exists. We
will see below that we cannot hold this assumption.
Now that we can represent sets, we want to compare them. We can simply define relations between
sets using the three set description operations introduced above.
Relations between Sets
set equality: A ≡ B :⇔ ∀x.x ∈ A ⇔ x ∈ B
subset: A ⊆ B :⇔ ∀x.x ∈ A ⇒ x ∈ B
proper subset: A ⊂ B :⇔ (A ⊆ B) ∧ (A ≡ B)
superset: A ⊇ B :⇔ ∀x.x ∈ B ⇒ x ∈ A
proper superset: A ⊃ B :⇔ (A ⊇ B) ∧ (A ≡ B)
©: Michael Kohlhase 46
We want to have some operations on sets that let us construct new sets from existing ones. Again,
can define them.
Operations on Sets
union: A ∪ B := {x | x ∈ A ∨ x ∈ B}
union over a collection: Let I be a set and Si a family of sets indexed by I, then
i∈I Si :=
{x | ∃i ∈ I.x ∈ Si}.
intersection: A ∩ B := {x | x ∈ A ∧ x ∈ B}
intersection over a collection: Let I be a set and Si a family of sets indexed by I, then
i∈I Si := {x | ∀i ∈ I.x ∈ Si}.
set difference: A\B := {x | x ∈ A ∧ x ∈ B}
the power set: P(A) := {S | S ⊆ A}
the empty set: ∀x.x ∈ ∅
Cartesian product: A × B := {〈a, b〉 | a ∈ A ∧ b ∈ B}, call 〈a, b〉 pair.