Assign Yourself and Do NowThursday, January 10, 2013

Do Now Explanation

Truth Table

p ¬p p∨¬p

T F TF T T

Explanation

• It will always be true – since OR means at least one, and they are opposites, one of them will be true always.

Truth Tables

Negation

“not”

Conjunction

“and”

Disjunction

“or”

Exclusive Disjunction“or, not both”

p q ¬p ¬q ∧ ∨

T T F F T T F

T F F T F T T

F T T F F T T

F F T T F F F

Sponge Bob and Patrick

In words, p v ¬q:

Sponge Bob lives under the sea or Sponge Bob and Patrick are not friends.

Truth Table

p q ¬q p v ¬q

T T F T

T F T T

F T F F

F F T T

Under what conditions is p v ¬q true?

• When Sponge Bob lives under the sea

• When Spongebob and Patrick are not friends

• Both

New Definitions

Tautology

A compound proposition is a tautology if all the values in its truth table column are true.

Logical Contradiction

A compound proposition is a logical contradiction if all the values in its truth table column are false.

Determine if p v ¬p is a tautology, a logical contradiction or neither

p v ¬p – truth table

p ¬p p v ¬p

T F T

F T T

Conclusion?

It is a tautology because all the values in the p v ¬p column are TRUE.

Tautology, Logical Contradiction or Neither?

p q p ^ q p v q ¬ (p v q) (p ^ q) ^ ¬ (p v q)

T T T T F F

T F F T F F

F T F T F F

F F F F T F

(p ^ q) ^ ¬ (p v q) is a logical contradiction because all of the values in its column are false.

¬(p^q)

Meaning in Words

¬(p^q) =

¬(Brittany likes volleyball and math) =

Brittany does not like both volleyball and math (she dislikes at least one).

Truth Tablep q p ^ q ¬(p^q)

T T T F

T F F T

F T F T

F F F T

¬p v ¬q

Meaning in Words

¬p v ¬q =

Brittany does not like volleyball or Brittany does not like math (or both).

This is neither a tautology nor a logical contradiction because the last column is not purely T or F.

Truth Tablep q ¬p ¬q ¬p v ¬q

T T F F F

T F F T T

F T T F T

F F T T T

Compare the Two!

¬ p v ¬ q Truth Tablep q ¬p ¬q ¬p v ¬q

T T F F F

T F F T T

F T T F T

F F T T T

¬ (p ^ q) Truth Tablep q p ^ q ¬(p^q)

T T T F

T F F T

F T F T

F F F T

If two truth tables have the same end result, then the two statements are logically equivalent.

Try the Lizzy Truth Table

1. Make your columns: p, q, r, ¬ r, p v q, (p v q) ^ ¬ r

2. The IB will help you by making the table the right size

3. Because we have three original propositions (p, q, r), we will have 23 = 8 rows below the header.

Lizzy Truth Table… Finish the Rest!

p q r ¬ r p v q (p v q) ^ ¬ r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Lizzy Truth Table Answer

p q r ¬ r p v q (p v q) ^ ¬ r

T T T F T F

T T F T T T

T F T F T F

T F F T T T

F T T F T F

F T F T T T

F F T F F F

F F F T F F

For Tomorrow’s Quiz

You should be able to:

1. Say if something is/isn’t a proposition. (Tues.)

2. Negate propositions. (Tues.)

3. Use conjunctions (and, ^), disjunctions (at least one, v), exclusive disjunctions (either/or, v). (Wed.)

4. Say if a statement is a tautology, logical contradiction, or neither. (Thurs.)

5. Say if two statements are logically equivalent. (Thurs.)

HW Check/ Time For HW

• P. 540, #1, 2, 3, 4, 6, 8• P. 542 # 1, 2, 3, 4, 5, 6

do a and b. If there is more than one sub question, do i & ii