| CS 420 |
| 2024fall |
| ibarland |
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Sets and set-notation; functions and relations (1-1, onto); cardinalities
- logic notation: translate engl. <-> logic:
- ∃m∈ℕ.n+m=0 "n has an additive-inverse in ℕ" [note: n is a free var.]
This formula is true if n happens to be 0 or -5, but false if n=17
- ∀n∈ℕ.∃m∈ℕ.n+m=0 "every number in ℕ has an additive inverse in ℕ" [note: NO free var]
False.
- if we replace the previous with "m∈ℤ", it becomes true.
- "n/2 is an integer" (a.k.a. "n is even")
n/2 ∈ ℤ
Now, write it w/o referring to division (but mult. okay)
{{{ thinking: n = 2*(some integer). NOW we can logic-ize it!: }}}
∃k∈ℤ. n=2*k
Thus we could define the set of even numbers as
2ℤ = { n | ∃k∈ℤ. n=2*k }
- ∃k∈N. n=5k in english is: "n is divisible by 5"
- "The set ℤ is closed under squaring" (which is true, btw)
∀z∈ℤ. (z²∈ℤ)
- a natnum n is composite ("non-prime",nearly):
(n∈ℕ) ∧ (∃k∈ℕ .∃m∈ℕ. n=m*k ^ n>1 ^ m>1)
(n∈ℕ) ∧ (∃k,m∈ℕ. n=m*k, n,m>1)
- "every prime# is odd" (which, btw, is false);
use an `->` to capture "for all primes" rather than "for all integers":
forall n, prime(n) -> n is odd
∀n. prime(n) → ∃k.n=2k+1
∀n. prime(n) → ∀k.n≠2k
where prime(n) is shorthand for: ((n∈ℕ) ∧ ¬(∃k∈ℕ .∃m∈ℕ. n=m*k ^ n>1 ^ m>1) ∧ n≠1)
How can we further modify this, to make it true over N?
- if n a mult. of 5, then n ends in '0' or '5' (use function 'lastDigitOfNumeral_10')
- f is an inverse of g
(Actually, note: we have "left-inverse" and "right-inverse". Sometimes g is both of those; sometimes not.)
- relation R A×B is a function
recall: a relation is just a set of pairs: e.g.
{ (16,4), (9, 3), (256,16), (16,-4), … }
a relation is a *function* iff: ("there is only one output for a given input")
∀ a∈A: ∀b,c. (a,b)∈R ^ (a,c)∈R → b=c.
∧ ∃z.(a,z)∈R
- every 'a' is followed by a 'b'; use "charAt"
- string w has string p as a prefix
- book 4.3 (p.35+15=50): for a language L over an alphabet Σ
From book: given a language L over Σ:
chop(L)={w: ∃x∈L (x=x₁cx₂ ∧ x₁∈Σ* ∧ x₂∈Σ* ∧ c∈Σ ∧ |x₁|=|x₂| ∧ w=x₁x₂) }
wtf?? Let's work it out!
firstchars(L) = (see book for def'n, and we'll work through it)
Next:
- Writing FSMs (Chpt.05)
- Example 5.2
- Example 5.3, every "a" region in w is even-length:
- Task: odd-parity (soln: Example 5.4)
Now back to formalizing:
Def'n: FSA (as tuple), p.42+15
- define configuration
- define acceptance
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