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lect01b
propositions; truth-tables
- logic: a language for writing proofs (and, specs)
propositions: the vocabulary of our language.
http://cnx.org/content/m10715/latest/
(HoP example)
- connectives: and/or/not;implies;iff
or: "lunch money or else"; "and/or"
if: "If you commit a crime, then you go to jail."
"If my baby is a girl, I'll name her Jessie"
"If you know Paris Hilton's phone#, then I'm a mule's uncle!"
"the receiver gained 20yds if he gained an inch!"
(but perhaps s/he caught it out of bounds, upon replay)
aka: "my candidate got 10000 votes if they got one"
iff: crime iff jail;
- work through Book: #1,5,11.
Q: T or F: "if today is Wednesday, we are skipping lecture"
Suppose "if they like me, they'll call me tonight" is true.
Suppose you get called. Can you conclude they like you?
No: "F -> T" and "T -> T" are both T.
Suppose you don't get called. Can you conclude they don't like you?
Yes: if q is false and yet "p->q" is true, then p must be false.
- make a truth table for...
p->q, and for -p v q, and for q->p, and for -q -> -p.
Note: they are *equivalent*. We write p->q ≡ -p v q.
Are the following two equivalent?
(distributive law)
p v (q ^ r) (pvq) ^ (pvr)
p^F; p^T
p->q <-> (-q->-p)
p v -p
"tautology"
a->(b->c) vs (a->b)->c
- bitstrings (and,or on bitstrings)
=====What about Boolean alg?!
Example 1: contrapositive:
a -> b
≡ -a v b (by def'n of ->)
≡ b v -a (by commutativity of v)
≡ --b v -a (by double-complementation)
≡ -b -> -a (by def'n of ->, with phi = -b, and psi = -a)
This last step is a bit subtle.
Example 2:
Show the absorption law, W/O using it itself.
The very first step is devious!:
a ^ (avb)
≡ (a v F) ^ (a v b) (by idempotence of ∨ over F)
≡ a v (F ^ b) (by distribution of v over ^, in reverse)
≡ a v F (by dominance of F over ^)
≡ a (by idempotence of ∨ over F)
Note: we've shown that the absorption law is actually redundant.
It makes you wonder: what is a *minimal* set of equivalencies?!
Def'n: CNF:
A formula is in "Conjunctive Normal Form" if it is written as
phi_1 ^ phi_2 ^ ... ^ phi_n
where each phi_i is written as
(l_1 v l_2 v ... v l_k)
where each l_j is either a proposition or the negation of a proposition.
Examples:
(a v b v -c) ^ (a v -b v c) ^ (a v b v c)
Is there a simpler example? Yes:
(a v b) ^ (a v -c)
An even simpler example? Yes:
(a) ^ (b) [here we have two clauses, each w/ one term]
(a v b) [one clause with two terms]
An even simpler example? Yes:
a
An even simpler example?
Hmm-- can we make the conjunction of zero clauses?
The empty-formula is ambiguous, but "T" would be the logical
value of conjunction-of-zero-things.
We need a definition of what a formula is, to decide this.
(Most definitions dis-allow the empty formula.
However, the conjunction of zero things is True,
so perhaps we'd consider "T" in CNF???)
Non-examples:
(a ^ b ^ c) v (a ^ -b) [this is actually in *disjunctive* normal form!]
(a v (b ^ (c v a)))
a -> b
a v (b ^ c)
Non-non-examples:
(a v b) [that's already listed above]
a ^ b [also listed above]
Note: don't confuse "a formula in CNF" with
"a formula which is equivalent to some *other* formula in CNF".
E.g., we've seen "a -> b" is equivalant to "(-a v b)" which is
a one-clause CNF formula.
Do you think *every* formula is equivalent to CNF?
WHy or why not?
Who cares about normal forms?
Consider to equivlanet *algebra* formulas (equations):
x^3 - x = 0
and equivalently
x(x+1)(x-1)=0.
In the second version, it's easy to read off all the
values of x which make the formula true.
In the first version, it wasn't obvious.
That's why we prefer the first version.
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