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lect05c
proofs; sets
Set:
- def'n. No order, no repetition.
Define explicitly, or as a rule (set-builder notation)
- common sets: φ, N, Z, Q, R, C; B, Sigma, Sigma*
intervals of real numbers: [a,b] (a,b)
eg [0,1]; (0,Infinity); [0,1).
- def'n 'subset'; equals. Proof A = B iff A⊆B, B⊆A.
- constructors (w/ Venn Diagrams)
union, intersect, complement, set-diff (R-Q; N-{0})
Examples: P=primes, O=odd, E=even.
E ∪ O = ??
E ∩ O = ??
P ∩ E = ?
P ∪ O = ?
cart.prod (People x Color, People x People, ...)
Color: [0,256)x[0,256)x[0,256)
You know, in retrospect, Java *kinda* has some limited union,intersection:
use interfaces.
- computer representation:
bitmap (for small finite lists -- in a mario save-file, which stars found)
Big-Union:
Consider M_2 = {0,2,4,...}, M_3 = {0,3,6,9,...}.
Give closed-form def'n.
Consider Union_i″Primes M_i.
What is a simpler way of expressing this set?
Russel's paradox:
U - universe. Is U″U?
B - "Bertie": {x | x″ x }. is B″B?
R - "Russel": {x | ¬(x″x) }. is R″R?
Functions:
f : A -> B
domain, codomain.
Example:
sqrt;
string-length
favorite-color : Person -> Color
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