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# CS322 - PowerPoint PPT Presentation

Week 1 - Wednesday. CS322. Last time. Course overview Propositional logic Truth tables AND, OR, NOT Logical equivalence. Questions?. Logical warmup. You come to a fork in the road Two men stand beneath a sign that reads: Ask for the way, but waste not your breath

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### CS322

• Course overview

• Propositional logic

• Truth tables

• AND, OR, NOT

• Logical equivalence

Logical warmup

• You come to a fork in the road

• Two men stand beneath a sign that reads:

• One road is freedom, the other is death

• Just one of the pair will lead you aright

• For one is a Knave, the other a Knight

• What single yes or no question can you ask to determine which fork to take?

• What’s an expression that logically equivalent to ~(p  q) ?

• What about logically equivalent to ~(p  q) ?

• De Morgan’s Laws state:

• ~(p  q)  ~p ~q

• ~(p  q)  ~p ~q

• Essentially, the negation flips an AND to an OR and vice versa

• A tautology is something that is true no matter what

• Examples:

• T

• p ~p

• pp

• The final column in a truth table for a tautology is all true values

• The book sometimes writes a statement which is a tautology as a t

• A contradiction is something that is false no matter what

• Examples:

• F

• p ~p

• ~(pp)

• The final column in a truth table for a contradiction is all false values

• The book sometimes writes a statement which is a contradiction as a c

• You can construct all possible outputs using combinations of AND, OR, and NOT

• But, sometimes it’s useful to introduce notation for common operations

• This truth table is for pq

• We use  to represent an if-then statement

• Let p be “The moon is made of green cheese”

• Thus, pq is how a logician would write:

• If the moon is made of green cheese, then the earth is made of rye bread

• Here, p is called the hypothesis and q is called the conclusion

• What other combination of p and q is logically equivalent to pq ?

• pq is true when:

• p is true and q is true

• p is false

• Why?

• For the whole implication to be true, the conclusion must always be true when the hypothesis is true

• If the hypothesis is false, it doesn’t matter what the conclusion is

• “If I punch the tooth fairy in the face, I will be Emperor of the World”

• What’s the negation of an implication?

• Given a conditional statement pq, its contrapositive is ~q ~p

• Conditional:

“If a murderer cuts off my head, then I will be dead.”

• Contrapositive:

“If I am not dead, then a murderer did not cut off my head.”

• What’s the relationship between a conditional and its contrapositive?

• Given a conditional statement pq:

• Its converse is qp

• Its inverse is ~p ~q

• Consider the statement:

• “If angry ham sandwiches explode, George Clooney will become immortal.”

• What is its converse?

• What is its inverse?

• How are they related?

• Sometimes people say “if and only if”, as in:

• “A number is prime if and only if it is divisible only by itself and 1.”

• This can be written piffq or pq

• This is called the biconditional and has this truth table:

• What is the biconditional logically equivalent to?

• 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

• Modus tollens is a contrapositive reworking of the argument, which is also valid:

• p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 q

• 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

• 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

•  p

• p q

• ~p

•  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

•  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

•  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

• 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?

• Predicate logic

• Universal quantifier

• Existential quantifier