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

y stating element-hood (a ∈ S) or not (b ∈ S).
Warning: Learn to distinguish between objects and their representations!
({a, b, c} and {b, a, a, c} are different representations of the same set)
c○: Michael Kohlhase 44
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 :⇔ (∀x.x ∈ A ⇒ x ∈ B) ∧ (A ≡ B)
superset: A ⊇ B :⇔ ∀x.x ∈ B ⇒ x ∈ A
proper superset: A ⊃ B :⇔ (∀x.x ∈ B ⇒ x ∈ A) ∧ (A ≡ B)
c○: Michael Kohlhase 45
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.
n-fold Cartesian product: A1 × · · · × An := {〈a1, . . . , an〉 | ∀i.(1 ≤ i ≤ n) ⇒ ai ∈ Ai},
call 〈a1, . . . , an〉 an n-tuple
n-dim Cartesian space: A n := {〈a1, . . . , an〉 | (1 ≤ i ≤ n) ⇒ ai ∈ A},
call 〈a1, . . . , an〉 a vector
Definition 48 We write n
i∈{i∈N | 1≤i≤n} Si.
i=1 Si for
i∈{i∈N | 1≤i≤n} Si and n i=1 Si for
c○: Michael Kohlhase 46
These operator definitions give us a chance to reflect on how we do definitions in mathematics.
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