Week 2 - Friday

 What did we talk about last time?  Predicate logic  Negation  Multiple quantifiers

 There are two lengths of rope  Each one takes exactly one hour to burn completely  The ropes are not the same lengths as each other  Neither rope burns at a consistent speed (10% of a rope could take 90% of the burn time, etc.)  How can you burn the ropes to measure out exactly 45 minutes of time?

 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"

 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

 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)

 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.”

 “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”   y  D,  x  P such that S(x,y)   x  P,  y  D such that S(x,y)

 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

 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.

 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

 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

 Quantification adds new features to an argument  The most fundamental is universal instantiation  If a property is true for everything in a domain (universal quantifier), it is true for any specific thing in the domain  Example:  All the party people in the place to be are throwing their hands in the air!  Julio is a party person in the place to be   Julio is throwing his hands in the air

 Formally,   x, P(x)  Q(x)  P(a) for some particular a Q(a)Q(a)  Example:  If any person disses Dr. Dre, he or she disses him or herself  Tammy disses Dr. Dre  Therefore, Tammy disses herself

 Much the same as universal modus ponens  Formally,   x, P(x)  Q(x)  ~Q(a) for some particular a   ~P(a)  Example:  Every true DJ can skratch  Carl Strikwerda can’t skratch  Therefore, Carl Strikwerda is not a true DJ

 Unsurprisingly, the inverse and the converse of universal conditional statements do not have the same truth value as the original  Thus, the following are not valid arguments:  If a person is a superhero, he or she can fly.  Astronaut John Blaha can fly.  Therefore, John Blaha is a superhero. FALLACY  A good man is hard to find.  Osama Bin Laden is not a good man.  Therefore, Osama Bin Laden is not hard to find. FALLACY

 We can test arguments using Venn diagrams  To do so, we draw diagrams for each premise and then try to combine the diagrams Touchable things This

 All integers are rational numbers  is not rational  Therefore, is not an integer rational numbers integers rational numbers integers

 All tigers are cats  Panthro is a cat  Therefore, Panthro is a tiger cats tigers cats Panthro cats tigers cats tigers Panthro

 Diagrams can be useful tools  However, they don’t offer the guarantees that pure logic does  Note that the previous slide makes the converse error unless you are very careful with your diagrams

 In Java (and many other languages), there are two kinds of equal signs: = (assignment) and == (testing for equality)  Mathematicians have two as well, but they use the same symbol!  One kind of equal sign is a definition  The other kind is a theorem

 The other kind of equals shows a fact that has been derived from other facts  We've discovered that it's true!  Example:  3x + 2 = 11  3x = 9  x = 3  Sometimes confusion arises when people mistake one kind of equals for another Definition Theorem

 Proofs and counterexamples  Basic number theory

 Assignment 1 is due tonight at midnight  Read Chapter 4  Start on Assignment 2  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 today 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