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Week 2 - Wednesday. CS322. Last time. What did we talk about last time? Arguments Digital logic circuits Predicate logic Universal quantifier Existential quantifier. Questions?. Logical warmup. 1. Four men are standing in front of a firing-squad #1 and #3 are wearing black hats - PowerPoint PPT Presentation
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CS322Week 2 - Wednesday
Last time
What did we talk about last time? Arguments Digital logic circuits Predicate logic
Universal quantifier Existential quantifier
Questions?
Logical warmup Four men are standing in front of a firing-
squad #1 and #3 are wearing black hats #2 and #4 are wearing white hats They are all facing the same direction with a
wall between #3 and #4 Thus,
#1 sees #2 and #3 #2 sees #3 #3 and #4 see no one
The men are told that two white hats and two black hats are being worn
The men can go if one man says what color hat he's wearing
No talking is allowed, with the exception of a man announcing what color hat he's wearing.
Are they set free? If so, how?
1
3
2
4
Digital Logic Review
Common gates The following gates have the same function as
the logical operators with the same names:
NOT gate:
AND gate:
OR gate:
Digital logic exercises
Build an OR circuit using only AND and NOT gates
Build a bidirectional implication circuit using AND, OR, and NOT gates
Predicate Logic
Universal quantification
The universal quantifier means “for all” The statement “All DJ’s are mad ill” can
be written more formally as:x D, M(x)
Where D is the set of DJ’s and M(x) denotes that x is mad ill
Notation: P(x) Q(x) means, for predicates P(x) and
Q(x) with domain D: x D, P(x) Q(x)
Existential quantification The universal quantifier means
“there exists” The statement “Some emcee can
bust a rhyme” can be written more formally as:
y E, B(y) Where E is the set of emcees and B(y)
denotes that y can bust a rhyme
Quantified examples Consider the following:
S(x) means that x is a square R(x) means that x is a rectangle H(x) means that x is a rhombus P is the set of all polygons
Which of the following is true: x P, S(x) R(x) x P, R(x) S(x) x P, R(x) H(x) S(x) x P, R(x) ~S(x) x P, ~R(x) H(x) x P, R(x) ~S(x) x P, ~H(x) S(x)
More quantified examples Convert the following statements in
English into quantified statements of predicate logic Every son is a descendant Every person is a son or a daughter There is someone who is not a
descendant Every parent is a son or a daughter There is a descendant who is not a son
Negating Quantifiers and Multiple QuantifiersStudent Lecture
Negating quantified statements When doing a negation, negate the
predicate and change the universal quantifier to existential or vice versa
Formally: ~(x, P(x)) x, ~P(x) ~(x, P(x)) x, ~P(x)
Thus, the negation of "Every dragon breathes fire" is "There is one dragon that does not breathe fire"
Negation example Argue the following:
"Every unicorn has five legs" First, let's write the statement formally
Let U(x) be "x is a unicorn" Let F(x) be "x has five legs" x, U(x) F(x)
Its negation is x, ~(U(x) F(x)) We can rewrite this as x, U(x) ~F(x)
Informally, this is "There is a unicorn which does not have five legs"
Clearly, this is false If the negation is false, the statement must be true
Vacuously true
The previous slide gives an example of a statement which is vacuously true
When we talk about "all things" and there's nothing there, we can say anything we want
Conditionals Recall:
Statement: p q Contrapositive:~q ~p Converse:q p Inverse: ~p ~q
These can be extended to universal statements: Statement: x, P(x) Q(x) Contrapositive:x, ~Q(x) ~P(x) Converse:x, Q(x) P(x) Inverse: x, ~P(x) ~Q(x)
Similar properties relating a statement equating a statement to its contrapositive (but not to its converse and inverse) apply
Necessary and sufficient
The ideas of necessary and sufficient are meaningful for universally quantified statements as well:
x, P(x) is a sufficient condition for Q(x) means x, P(x) Q(x)
x, P(x) is a necessary condition for Q(x) means x, Q(x) P(x)
Multiple Quantifiers
Multiple quantifiers
So far, we have not had too much trouble converting informal statements of predicate logic into formal statements and vice versa
Many statements with multiple quantifiers in formal statements can be ambiguous in English
Example: “There is a person supervising every
detail of the production process.”
Example
“There is a person supervising every detail of the production process.”
What are the two ways that this could be written formally? Let D be the set of all details of the production
process Let P be the set of all people Let S(x,y) mean “x supervises y”
x D, y P such that S(x,y)
y P, x D such that S(x,y)
Mechanics Intuitively, we imagine that corresponding
“actions” happen in the same order as the quantifiers
The action for x A is something like, “pick any x from A you want”
Since a “for all” must work on everything, it doesn’t matter which you pick
The action for y B is something like, “find some y from B”
Since a “there exists” only needs one to work, you should try to find the one that matches
Tarski’s World Example
Is the following statement true? “For all blue items x, there is a green item y with the
same shape.” Write the statement formally. Reverse the order of the quantifiers. Does its truth value
change?
a
c
g
b
d
f
i
k
e
h
j
Practice
Given the formal statements with multiple quantifiers for each of the following: There is someone for everyone. All roads lead to some city. Someone in this class is smarter than
everyone else. There is no largest prime number.
Negating multiply quantified statements The rules don’t change Simply switch every to and every
to Then negate the predicate Write the following formally:
“Every rose has a thorn” Now, negate the formal version Convert the formal version back to
informal
Changing quantifier order As show before, changing the order of
quantifiers can change the truth of the whole statement
However, it does not necessarily Furthermore, quantifiers of the same type
are commutative: You can reorder a sequence of quantifiers
however you want The same goes for Once they start overlapping, however, you can’t
be sure anymore
Quiz
Upcoming
Next time…
Arguments with quantifiers
Reminders Keep reading Chapter 3 Assignment 1 is due Friday at midnight The company EC Key is sponsoring a contest to
come up with novel uses for their BlueTooth door access technology Interested? Come to the meeting this Friday, 1/22 at
3:30pm in Hoover 110 Teams will be formed from CS, engineering, and
business students Ask me for more information!
Also, there's a field trip to Cargas Systems in Lancaster next Friday