CS322

1 / 36

# 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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' CS322' - viho

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Week 1 - Wednesday

### CS322

Last time
• 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?
De Morgan’s Laws
• 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
Tautology
• 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
What are you implying?
• 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
If…
• 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 ?
Why is implication used that way?
• 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?
Contrapositive
• 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?
Converse and inverse
• 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?
Biconditional
• 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?
Arguments
• 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
Examples
• 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)
Common argument tools
• 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
Generalization
• 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.”
Specialization
• 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.”
Conjunction
• 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.”
Elimination
• 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.”
Transitivity
• 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.”
Division into cases
• 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.”
Fallacies
• 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
Common fallacies
• 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
• 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
Common gates
• The following gates have the same function as the logical operators with the same names:
• NOT gate:
• AND gate:
• OR gate:
Example
• Draw the digital logic circuit corresponding to:

(p ~q)  ~(p  r)

• What’s the corresponding truth table?
Next time…
• Predicate logic
• Universal quantifier
• Existential quantifier
Reminders