Week 1 - Friday

 What did we talk about last time?  Tautologies  Contradictions  Laws of Boolean algebra  Implications  Inverses  Converses  Bidirectional

 For Diwali, Mr. Patel's five daughters gave each other books as presents.  Each presented four books and each received four books, but no two girls divided her books in the same way.  That is, only one gave two books to one sister and two to another. Bharat gave all her books to Abhilasha; Chandra gave three to Esha.  Who gave how many books to whom?

 One note about implications and wording them:  p is a sufficient condition for q means p  q  p is a necessary condition for q means q  p  This nomenclature is a touch counterintuitive  Think of it this way:  p  q means that p is enough to get you q, but there might be other things that will get you q  q  p means that, since you automatically get p when you've got q, there's no way to have q without p

 An argument is a list of statements (called premises) followed by a single statement (called a conclusion)  Whenever all of the premises are true, the conclusion must also be true, in order to make the argument valid

 Are the following arguments valid?  p  q  ~r(premise)  q  p  r(premise)   p  q(conclusion)  p  (q  r)(premise)  ~r(premise)   p  q(conclusion)

 Modus ponens is a valid argument of the following form: p  qp  q pp  q q  Modus tollens is a contrapositive reworking of the argument, which is also valid: p  qp  q ~q~q   ~p  Give verbal examples of each  We call these short valid arguments rules of inference

 The following are also valid rules of inference: pp  p q p q qq  p q p q  English example: “If pigs can fly, then pigs can fly or swans can breakdance.”

 The following are also valid rules of inference:  p  q  p p  q q  English example: “If the beat is out of control and the bassline just won’t stop, then the beat is out of control.”

 The following is also a valid rule of inference: pp qq   p  q  English example: “If the beat is out of control and the bassline just won’t stop, then the beat is out of control and the bassline just won’t stop.”

 The following are also valid rules of inference:  p  q ~q~q  p p ~p~p  q q  English example: “If you’re playing it cool or I’m maxing and relaxing, and you’re not playing it cool, then I’m maxing and relaxing.”

 The following is also a valid rule of inference:  p  q q  rq  r   p  r  English example: “If you call my mom ugly I will call my brother, and if I call my brother he will beat you up, then if you call my mom ugly my brother will beat you up.”

 The following is also a valid rule of inference:  p  q  p  r q  rq  r  r r  English example: “If am fat or sassy, and being fat implies that I will give you trouble, and being sassy implies that I will give you trouble, then I will give you trouble.”

 The following is also a valid rule of inference:  ~p  c  p p  English example: “If my water is at absolute zero then the universe does not exist, thus my water must not be at absolute zero.”

 A fallacy is an argument that is not valid  It could mean that the conclusion is not true in only a single case in the truth table  But, if the conclusion is ever false whenever all the premises are true, the argument is a fallacy  Most arguments presented by politicians are fallacies for one reason or another

 Converse error  If Joe sings a sad song, then Joe will make it better.  Joes makes it better.  Conclusion: Joe sings a sad song. FALLACY  Inverse error  If you eat too much, you will get sick.  You are not eating too much.  Conclusion: You will not get sick. FALLACY

 Digital logic circuits are the foundation of all computer hardware  Circuits are built out of components called gates  A gate has one or more inputs and an output  Gates model Boolean operations  Usually, in digital logic, we use a 1 for true and a 0 for false

 The following gates have the same function as the logical operators with the same names:  NOT gate:  AND gate:  OR gate:

 Draw the digital logic circuit corresponding to: (p  ~q)  ~(p  r)  What’s the corresponding truth table?

 A predicate is a sentence with a fixed number of variables that becomes a statement when specific values are substituted for to the variables  The domain gives all the possible values that can be substituted  The truth set of a predicate P(x) are those elements of the domain that make P(x) true when they are substituted

 Let P(x) be “x has had 4 wisdom teeth removed”  What is the truth set if the domain is the people in this classroom?  Let Q(n) be “n is divisible by exactly itself and 1”  What is the truth set if the domain is the set of positive integers Z + ?

 We will frequently be referring to various sets of numbers in this class  Some typical notation used for these sets:  Some authors use Z + to refer to non-negative integers and only N for the natural numbers SymbolSetExamples RReal numbersVirtually everything that isn’t imaginary ZIntegers{…, -2, -1, 0, 1, 2,…} Z-Z- Negative integers{-1, -2, -3, …} Z+Z+ Positive integers{1, 2, 3, …} NNatural numbers{1, 2, 3, …} QRational numbers a/b where a,b  Z and b  0

 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

 Let S = {1, 2, 3, 4, 5}  Show that the following statement is true:   x  S, x 2 ≥ x  Show that the following statement is false:   x  R, x 2 ≥ x

 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

 Let S = {2, 4, 6, 8}  Show that the following statement is false:   x  S, 1/x = x  Show that the following statement is true:   x  Z, 1/x = x

 Tarski’s World provides an easy framework for testing knowledge of quantifiers  The following notation is used:  Triangle(x) means “x is a triangle”  Blue(y) means “y is blue”  RightOf(x, y) means “x is to the right of y (but not necessarily on the same row)”

 Are the following statements true or false?   t, Triangle(t)  Blue(t)   x, Blue(x)  Triangle(x)   y such that Square(y)  RightOf(d, y)   z such that Square(z)  Gray(z) a a c c g g b b d d f f i i k k e e h h j j

 Negating quantifications  Multiple quantifications

 Keep reading Chapter 3  Start working on Assignment 1  Due next Friday