Transcript 11.3: Uses and Abuses of tests

Symbol
p, q, or r
¬
∧
∨
∨
In Words
Meaning
What
is
Logic?
 Logic is a way to describe situations or knowledge that
enables us to reason from existing knowledge to new
conclusions.
 For example:
 All college students are poor
 I am college a student
 Therefore, I am poor
 If the original statement is false, the conclusion is still
logical (it just may be false)
 All college students are rich (not true)
 I am a college student
 Therefore, I am rich!
Logical Reasoning
 You have four cards. A letter A – Z is the blue side.
 A number 0 – 9 is on the green side.
 You have these cards:
 What card do you turn over to test the rule “If a card
has a vowel on the blue side, it must have an even
number on the greenside?
Sets and Logical Reasoning
 A proposition is a statement that may be true or false.
 For Example,
 “Mrs. Skaff is a math teacher” is a true proposition
 “Mrs. Skaff is 10 feet tall” is a false proposition
 “Today is Saturday” is indeterminate because it may be
true of false depending on the circumstances.
 All of these are examples of simple propositions.
Propositions
 Consider the following statements:
 Go get the book
 Have you seen my new shirt?
 The dog is behind the shed
 Which statement is a proposition?
 The dog is behind the shed.
 Rewrite the others so they are propositions
 Siya got the book.
 Jackson saw my new shirt.
Representing propositions
 We represent propositions by letters such as p, q, and r.
 For Example:
 p: Mrs. Skaff is a math teacher
 q: Mrs. Skaff is 10ft tall
 r: Today is Sunday
 Opposites are represented by negation (¬)
 ¬p represents the opposite of p
 We read that as “not p”



¬p: Mrs. Skaff is not a math teacher
¬q: Mrs. Skaff is not 10ft tall
¬r: Today is not Sunday
Compound Propositions
 Compound propositions are statements which are
joined using and or or.
 Conjunctions
 When two propositions are joined using and the new
proposition is the conjunction of the originals.
 The conjunction of the two propositions p and q is
denoted by p∧q
Truth Values
 Because a proposition is a statement that can be either
true or false, its Truth Values are T for true and F for
false. This may be represented in a table.
p : Carla has
black hair
T
F
¬ p : Carla does
not have black
hair
Conjunctions
 For the following propositions:
 p: Ty is a superhero
 q: Sofia is a basketball star
 What is p^q :
 Ty is a superhero and Sofia is a basketball star
 What is ¬p^q
 Ty is not a superhero and Sofia is a basketball
star
Conjunction: Truth Value
 The truth value of a conjunction is only true with
BOTH the propositions are true.
p
T
T
F
F
q
T
F
T
F
p∧q
Conjunctions: Truth Value
p: Mrs. Skaff is a math teacher
q: Mrs. Skaff is 10ft tall
r: Today is Sunday
 Find the truth value of p∧q
p
q
p∧q
T
T
T
T
F
F
F
T
F
F
F
F
Simple Truth Table
p
q
¬q
p∧¬q
T
T
F
F
T
F
T
T
F
T
F
F
F
F
T
F
• Find the truth value of p^¬q
Disjunctions
 A disjunction is formed when propositions
are joined using the word “or”
 This is written as p ∨ q
 The truth value of a disjunction is true when at least
one of the propositions is true.
p
q
p∨q
T
T
T
T
F
T
F
T
T
F
F
F
Disjunction
 Example: Find the disjunction between the two
propositions
 p: Pikachu is a Pokemon
 q: Mrs. Dogancay is a Pokemon
 p is true (Mrs. Skaff’s favorite pokemon!)
 q is not true (unless there is something Mrs. Dogancay
is hiding from us)
 THEREFORE, p ∨ q is true.
p
q
p∨q
T
T
T
T
F
T
F
T
T
F
F
F
Exclusive
Disjunction
 Chandler will go home if John is running late or if it is
raining.
 You will go to Hawaii by boat or by plane.
 In the first, Chandler will go home if John is late, it’s
raning, or both
 In the second, you can travel by boat or plane, but not
both at the same time.
 Thi is called an exclusive disjunction
 p∨q
p
q
p∨q
T
T
F
T
F
T
F
T
T
F
F
F
Venn Diagrams
 You will go to Hawaii by boat or by plane
 p: you will go by boat
 q: you will go by plane
boat
plane
Practice
 p: London is the capital of England
q: 45 – 5 = 4
r: Cows have 4 legs
 Find the truth value of the following:
 p^q
 qvr
 ¬q ^ r
 ¬(p v q)
 pvr