Material Taken From:

Mathematics

for the international student

Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, 2004

Chapter 15A - Propositions

Mathematical Logic

Converting worded statements into symbols, then applying rules of deduction.

Example of deductive reasoning: • All teachers are poor.

• I am a teacher.

• By using logic, it follows that I am poor.

• • Logic, unlike natural language, is precise and exact.

Logic is useful in computers and artificial intelligence where there is a need to represent the problems we wish to solve using symbolic language.

BrainPop – Binary Video

For each of these statements, list the students for which the statement is true: a) I am wearing a green shirt.

b) I am not wearing a green shirt c) I am wearing a green shirt and green pants.

d) I am wearing a green shirt or green pants.

e) I am wearing a green shirt or green pants, but not both.

Propositions

Statements which may be true or false.

• • • • • Page 496 in the text.

Questions are not propositions.

Comments or opinions are not propositions. • Example: ‘Green is a nice color’ is subjective; it is not definitely true or false.

Propositions may be

indeterminate.

• Example: ‘your father is 177 cm tall’ would not have the same answer (true or false) for all people.

The

truth value

of a proposition is whether it is true or false.

Example 1

Which of the following statements are propositions? If they are propositions, are they true, false, or indeterminate?

a) 20  4 = 80 b) 25 × 8 = 200 c) Where is my pen?

d) Your eyes are blue.

Notation

• • We represent propositions by letters such as

p

,

q

and

r.

For example: –

p

: It always rains on Tuesdays.

–

q

: 37 + 9 = 46 –

r

:

x

is an even number.

Negation

• • • The negation of a proposition

p

as ¬

p

.

The truth value of ¬

p p

.

is “

not p

” and is written is the opposite of the truth value of For example: 

p

: It is Monday.

 ¬

p:

It is not Monday.

 

q:

Tim has brown hair.

¬

q:

Tim does not have brown hair.

Truth Tables

• Using the example: –

p

: It is Monday.

– ¬

p:

It is not Monday.

p

T F

¬p

F T

¬(¬p)

T F

Example 2

Find the negation of:

a)

x

is a dog for

x

 {dogs, cats} b)

x

≥ 2 for

x

 N c)

x

≥ 2 for

x

 Z

Section 15B - Compound Propositions Compound propositions S

tatements which are formed using ‘and’ or ‘or.’ • ‘and’ – 

conjunction

notation: p  q • ‘or’  –

disjunction

notation: p  q

Conjunction vs. Disjunction Examples

Conjunction

p

: Eli had soup for lunch

q

: Eli had a pie for lunch.

Disjunction

p

: Frank played tennis today

q

: Frank played golf today.

p



q

:

p



q

: •

p



q

is only true if both original propositions are true.

•

p



q

is true if one or both propositions are true.

•

p



q

is false only if both propositions are false.

Conjunction/Disjunction and Truth Tables

p

T T F F

q

T F T F

p



q p



q

T F F T T T F F

U

Conjunction/Disjunction and Venn Diagrams

Suppose P is the truth set of p, and Q is the truth set of q.

P



Q

the truth set for p  q is P  Q P Q the truth set for p  q is P  Q

P



Q

Examples 3 and 4

Write p  q for the following :

p

: Kim has brown hair,

q

: Kim has blue eyes Determine whether p  q is true or false:

p

: A square has four sides,

q

: A triangle has five sides

Examples 5 and 6

Write the disjunction p  q for

p

:

x

is a multiple of 2,

q

:

x

is a multiple of 5.

Determine whether p  q is true or false

p

: There are 100  in a right angle,

q

: There are 180  on a straight line.

Exclusive Disjunction

Is true when only one of the propositions is true.

• • notation: means “

p

or

q

, but not both” • For example, –

p

: Sally ate cereal for breakfast –

q

: Sally ate toast for breakfast

p

T T F F

q

T F T F

p



q

F T T F

Exclusive Disjunction

• In Logic ‘or’ is usually given in the inclusive sense.

– “

p

or

q

or both” • If the exclusive disjunction is meant, then it’ll be stated. – “

p

or

q

, but not both’ or “exactly one of

p

or

q

”

Example 7

Write the exclusive disjunction for

p

: Ann will invite Kate to her party,

q

: Ann will invite Tracy to her party.

Examples 8 and 9

• Consider

r: Kelly is a good driver

, and

s: Kelly has a good car.

Write in symbolic form: a) Kelly is a good driver and has a good car.

b) Kelly is not a good driver or has a good car.

• Consider

x: Sergio would like to go swimming tomorrow

, and

y: Sergio would like to go bowling tomorrow

Write in symbolic form: – Sergio would not like to go both swimming and bowling tomorrow.

Example 10

Define appropriate propositions and then write in symbolic form:

– Phillip likes ice cream or Phillip does not like Jell-O –

Homework (from 2

nd

edition)

• • • 17A.1 (every other problem) • #1, #2, #4, #5 17B.1 (every other problem) • #1, #2 17B.2

• #1ac, #2ad, #3a, #6ace, #7aeg, #11