1 Propositions A proposition is a declarative sentence that is either true or false Examples of propositions The Moon is made of green cheese Trenton is the capital of New Jersey Toronto is the capital of Canada ID: 260809 Download Presentation

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1. Propositions. A . proposition. is a declarative sentence that is either true or false.. Examples of propositions:. The Moon is made of green cheese.. Trenton is the capital of New Jersey.. Toronto is the capital of Canada..

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Slide1

Propositional Logic

1Slide2

Propositions

A

proposition

is a declarative sentence that is either true or false.

Examples of propositions:The Moon is made of green cheese.Trenton is the capital of New Jersey.Toronto is the capital of Canada.1 + 0 = 10 + 0 = 2Examples that are not propositions.Sit down!What time is it?x + 1 = 2x + y = z

2Slide3

Propositional Logic (or Calculus)

Constructing Propositions

Propositional

Variables

: p, q, r, s, …The proposition that is always true is denoted by T and the proposition that is always false is denoted by F.Compound Propositions: constructed from other propositions using logical connectivesNegation ¬Conjunction ∧Disjunction ∨Implication →Biconditional ↔

3Slide4

Compound Propositions: Negation

The

negation

of a proposition

p is denoted by ¬p and has this truth table:Example: If p denotes “The earth is round” then ¬p denotes “It is not the case that the earth is round,” or more simply “The earth is not round.” p

¬

p

T

F

F

T

4Slide5

Conjunction

The

conjunction

of propositions

p and q is denoted by p ∧ q and has this truth table:Example: If p denotes “I am at home” and q denotes “It is raining” then p ∧q denotes “I am at home and it is raining.”

p

q

p

∧

q

T

T

T

T

F

FFTFFFF

5Slide6

Disjunction

The

disjunction

of propositions

p and q is denoted by p ∨q and has this truth table:Example: If p denotes “I am at home” and q denotes “It is raining” then p ∨q denotes “I am at home or it is raining.”

p

q

p

∨

q

T

T

T

T

FTFTTFF

F

6Slide7

The Connective Or in English

In English “or” has two distinct meanings.

Inclusive

Or: For p ∨q to be T, either p or q or both must be T Example: “CS202 or Math120 may be taken as a prerequisite.”

Meaning: take either one or both

Exclusive

Or

(

Xor

). In p⊕q ,

either

p

or q but not both must be TExample: “Soup or salad comes with this entrée.” Meaning: do not expect to get both soup and salad

p

q

p

⊕

q

T

T

FTFTFTTFFF

7Slide8

Implication

If

p

and

q are propositions, then p →q is a conditional statement or implication, which is read as “if p, then q ”p is the hypothesis (antecedent or premise) and q is the

conclusion

(or

consequence

).

Example

: If

p

denotes “It

is

raining” and

q denotes “The streets are wet” then p →q denotes “If it is raining then streets are wet.”p

q

p

→

q

T

T

TTFFFTTFFT8Slide9

Understanding Implication

In

p

→

q there does not need to be any connection between p and q. The meaning of p →q depends only on the truth values of p and q. Examples of

valid

but

counterintuitive

implications:

“If the moon is made of green cheese, then you have more money than Bill Gates” --

True

“If Juan has a smartphone, then 2 + 3 = 6” -- False if Juan does have a smartphone, True if he does NOT

9Slide10

Understanding Implication

View logical conditional as an

obligation

or

contract:“If I am elected, then I will lower taxes”“If you get 100% on the final, then you will earn an A”If the politician is elected and does not lower taxes, then he or she has broken the campaign pledge. Similarly for the professor. This corresponds to the case where p is T and q is F. 10Slide11

Different Ways of Expressing

p

→

q

if p, then qif p, qp implies q q if p

q

when

p

q

whenever pNote

:

(

p

only if q) ≡ (q if p) ≢ (q only if p)It is raining

→

streets

are wet:

T

It is raining

only if

streets are wet: T

Streets are wet if it is raining: TStreets are wet only if it’s raining: F11

q

follows from

p

q

unless

¬p

p

only if

q

p

is sufficient for

q

q

is necessary for

pSlide12

Sufficient versus Necessary

Example 1:

p

→

q : get 100% on final → earn A in classp is sufficient for q: getting 100% on final is sufficient for earning A

q

is

necessary

for

p

:

earning A is necessary for getting 100% on final

Counterintuitive

in English:

“Necessary” suggests a precondition

Example is sequentialExample 2:Nobel laureate → is intelligentBeing intelligent is necessary for being Nobel laureateBeing a Nobel laureate is sufficient for being intelligent (better if expressed as: implies)

12Slide13

Converse, Inverse, and Contrapositive

From

p

→

q we can form new conditional statements .q →p is the converse of p →q ¬ p → ¬ q is the inverse of p →

q

¬

q

→ ¬

p

is the

contrapositive of p →

q

How are these statements related

to the original?

How are they related to each other?Are any of them equivalent?13Slide14

Converse, Inverse, and Contrapositive

Example

:

it’s raining

→ streets are wetconverse: streets are wet → it’s raininginverse: it’s not raining → streets are not wetcontrapositive: streets are not wet → it’s not rainingonly 3 is equivalent to original statement1

and

2

are not equivalent to original statement: streets could be wet for other reasons

1

and

2 are equivalent to

each other

14Slide15

Biconditional

If

p

and

q are propositions, then we can form the biconditional proposition p ↔q, read as “p if and only if q ”Example: If p denotes “You can take a flight” and q denotes “You buy a ticket” then

p

↔

q

denotes “You can take a

flight if and only if you buy a ticket”

True only if you do both or neitherDoing only one or the other makes the proposition false

p

q

p

↔q TTT

T

F

F

F

T

F

F

FT15Slide16

Expressing the Biconditional

Alternative ways to say “

p

if and only if

q”:p is necessary and sufficient for qif p then q, and converselyp iff q16Slide17

Compound Propositions

conjunction, disjunction, negation, conditionals, and

biconditionals

can be combined into arbitrarily complex

compound propositionAny proposition can become a term inside another propositionPropositions can be nested arbitrarilyExample: p, q, r, t are simple propositionsp ∨q , r ∧t , r →t are compound propositions using logical connectives(p ∨

q)

∧

t

and

(p

∨

q)

→

t are compound propositions formed by nesting17Slide18

Precedence of Logical Operators

Operator

Precedence

1 23 45

18

With multiple operators we need to know the

order

To reduce number of parentheses use precedence rules

Example

:

p

∨

q

→ ¬

r

is equivalent to

(

p

∨

q

) → ¬ r

If the intended meaning is

p

∨(

q

→ ¬

r

)

then parentheses must be usedSlide19

Truth Tables of Compound Propositions

Construction of a truth table:

Rows

One for every possible combination of values of all propositional variables

ColumnsOne for the compound proposition (usually at far right)One for each expression in the compound proposition as it is built up by nesting19Slide20

Example Truth Table

Construct a truth table for

p

q

rp qrp q →

r

T

T

T

T

F

F

T

T

F

TTTT FTTFFT

F

F

T

T

T

F

T

TTFFFTFTTTFFTFFTFFF

F

T

T

20

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