17 - smaris.edu.ecsmaris.edu.ec/wp-content/uploads/2013/08/logic.pdf · chapter 17 logic contents:...

17Chapter

Logic

Contents:

Syllabus reference: 3.4, 3.5, 3.6, 3.7

A

B

C

D

E

F

Propositions

Compound propositions

Truth tables and logicalequivalence

Implication andequivalence

Converse, inverse andcontrapositive

Valid arguments(extension)

IB_STSL-2edmagentacyan yellow black

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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_17\529IB_STSL-2_17.CDR Friday, 19 March 2010 3:24:13 PM PETER

OPENING PROBLEM

Brogan Padraig Sean

Things to think about:

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.

Mathematical logic deals with the conversion of worded

statements into symbols, and how we can apply rules of

deduction to them. The concept was originally suggested by

G W Leibniz (1646-1716).

George Boole (1815-1864) introduced the symbolism which

provided the tools for the analysis. Other people

who made important contributions to the field include

Bertrand Russell, Augustus DeMorgan, David Hilbert,

John Venn, Giuseppe Peano, and Gottlob Frege.

Mathematical arguments require basic definitions and

axioms, which are simple statements that we accept without

proof. Logical reasoning is then essential to building clear

rules based upon these definitions.

Propositions are statements which may be true or false.

Questions are not propositions.

Comments or opinions that are subjective, for example, ‘Green is a nice colour’ are also not

propositions since they are not definitely true or false.

Propositions may be indeterminate. For example, ‘your father is 177 cm tall’ would not

have the same answer (true or false) for all people.

PROPOSITIONSA

G W Leibniz

On Saint Patrick’s Day,

the students in a class are

all encouraged to wear

green clothes to school.

Eamonn

530 LOGIC (Chapter 17)


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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_17\530IB_STSL-2_17.CDR Friday, 12 February 2010 3:06:13 PM PETER

The truth value of a proposition is whether it is true or false.

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.

a This is a proposition. It is false as 20¥ 4 = 5.

b This is a proposition and is true.

c This is a question, so is not a proposition.

d This is a proposition. It is indeterminate, as the statement is true for some people

and false for other people.

NOTATION USED FOR PROPOSITIONS

We represent propositions by letters such as p, q and r.

For example, p: It always rains on Tuesdays.

q: 37 + 9 = 46r: x is an even number.

NEGATION

The negation of a proposition p is “not p”, and is written as :p.

The truth value of :p is the opposite of the truth value of p.

For example:

The negation of p: It is raining is :p: It is not raining.

The negation of p: Tim has brown hair is :p: Tim does not have brown hair.

From these examples, we can see that :p is

½false when p is true

true when p is false.

p :pT F

F T

This information can be represented in a truth table. The first

column contains the possible truth values for p, and the second column

contains the corresponding values for :p.

EXERCISE 17A.1

1 Which of the following statements are propositions? If they are propositions, are they

true, false or indeterminate?

a 11¡ 5 = 7 b 12 2fodd numbersgc 3

4 2 Q d 2 =2 Q

e A hexagon has 6 sides. f 37 2fprime numbersgg How tall are you? h All squares are rectangles.

i Is it snowing? j A rectangle is not a parallelogram.

Example 1 Self Tutor

531LOGIC (Chapter 17)


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k Your brother is 13. l Do you like dramatic movies?

m Joan sings loudly. n You were born in China.

o Alternate angles are equal. p Parallel lines eventually meet.

2 For each of the following propositions:

i write down the negation ii indicate if the proposition or its negation is true.

a p: All rectangles are parallelograms.

b m:p5 is an irrational number.

c r: 7 is a rational number.

d q: 23¡ 14 = 12

e r: 52¥ 4 = 13

f s: The difference between any two odd numbers is even.

g t: The product of consecutive integers is always even.

h u: All obtuse angles are equal.

i p: All trapeziums are parallelograms.

j q: If a triangle has two equal angles it is isosceles.

3 Find the negation of these propositions for x, y 2 R :

a x < 5 b x > 3 c y < 8 d y 6 10

Find the negation of:

a x is a dog for x 2 fdogs, catsgb x > 2 for x 2 N

c x > 2 for x 2 Z .

a x is a cat.

b x 2 f0, 1gc x < 2 for x 2 Z

5 Find the negation of:

a x > 5 for x 2 Z +

b x is a cow for x 2 fhorses, sheep, cows, goats, deergc x > 0 for x 2 Z

d x is a male student for x 2 fstudentsge x is a female student for x 2 ffemalesg.


4 For the two propositions r and s given:

i Is s the negation of r?

ii If s is not the negation of r, give the correct negation of r.

a r: Kania scored more than 60%. s: Kania scored less than 60%.

b r: Dea has less than two sisters. s: Dea has at least two sisters.

c r: Fari is at soccer practice. s: Fari is at music practice.

d r: I have been to Canada. s: I have never been to Canada.

e r:


I drank black tea today. s: I drank white tea today.


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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_17\532IB_STSL-2_17.CDR Tuesday, 9 March 2010 11:38:38 AM PETER

NEGATION AND VENN DIAGRAMS

Many propositions contain a variable.

The proposition may be true for some values of the variable, and false for others.

We can use a Venn diagram to represent these propositions and their negations.

For example, consider p: x is greater than 10.

U is the universal set of all the values that the

variable x may take.

P is the truth set of the proposition p, or the set

of values of x 2 U for which p is true.

P 0 is the truth set of :p.

For U = fx j 0 < x < 10, x 2 N g and proposition p: x is a prime number,

find the truth sets of p and :p.

If P is the truth set of p then P = f2, 3, 5, 7g.

The truth set of :p is P 0 = f1, 4, 6, 8, 9gThe Venn diagram representation is:

EXERCISE 17A.2

1 a Find the truth sets of these statements:

b Draw Venn diagrams to represent a i and a ii.

2 Suppose U = fstudents in Year 11g, M = fstudents who study musicg, and

O = fstudents who play in the orchestrag.

Draw a Venn diagram to represent the statements:

a All music students play in the school orchestra.

b None of the orchestral students study music.

c No-one in the orchestra does not study music.

3 a i Represent U = fx j 5 < x < 15, x 2 N g and p: x < 9 on a Venn diagram.

ii List the truth set of :p.

b i Represent U = fx jx < 10, x 2 N g and p: x is a multiple of 2on a Venn diagram.

ii List the truth set of :p.


U

P

P'

25

73

1

6 4

89

U

P

P'


i p: x is a multiple of 3, for U = fx j 20 < x < 30gii p: x is an even number, for U = fx j 1 < x 6 10giii p: x is a factor of 42, for U = Z .


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Compound propositions are statements which are formed using connectives

such as and and or.

CONJUNCTION

When two propositions are joined using the word and, the new proposition

is the conjunction of the original propositions.

If p and q are propositions, p ^ q is used to denote their conjunction.

For example, if p: Eli had soup for lunch

q: Eli had a pie for lunch

then p ^ q: Eli had soup and a pie for lunch.

We see that p ^ q is only true if Eli had both soup and a pie for lunch, which means that

both p and q are true.

If either of p or q is not true, or both p and q are not true, then p ^ q is not true.

A conjunction is true only when both original propositions are true.

The truth table below shows the conjunction “p and q”.

p q p ^ q

T T T

T F F

F T F

F F F

p ^ q is true when both p and q are true.

p ^ q is false whenever one or both of pand q are false.

The first 2 columns list the possible combinations for p and q.

We can use Venn diagrams to represent conjunctions.

Suppose P is the truth set of p, and

Q is the truth set of q.

The truth set of p ^ q is P \Q, the region where

both p and q are true.

EXERCISE 17B.1

1 Write p ^ q for the following pairs of propositions:

a p: Ted is a doctor, q: Shelly is a dentist.

b p: x is greater than 15, q: x is less than 30.

c p: It is windy, q: It is raining.

d p: Kim has brown hair, q: Kim has blue eyes.

COMPOUND PROPOSITIONSB

U

P Q

P Q\



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2 For the following pairs of propositions p and q, determine whether p^q is true or false:

a p: 5 is an odd number, q:

b p: A square has four sides, q:

c p: 39 < 27, q: 16 > 23

d p: 3 is a factor of 12, q: 4 is a factor of 12.

e p: 5 + 8 = 12, q: 6 + 9 = 15.

3 For U = fx j 1 6 x 6 12, x 2 Z g, consider the

propositions p: x is even and q: x is less than 7.

a Illustrate the truth sets for p and q on a Venn

diagram like the one alongside.

b Use your Venn diagram to find the truth set of

p ^ q.

DISJUNCTION

When two propositions are joined by the word or, the new proposition is the disjunction

of the original propositions.

If p and q are propositions, p _ q is used to denote their disjunction.

For example, if p: Frank played tennis today

and q: Frank played golf today

then p _ q: Frank played tennis or golf today.

p _ q is true if Frank played tennis or golf or both today.

So, p _ q is true if p or q or both are true.

A disjunction is true when one or both propositions are true.

Alternatively, we can say that

A disjunction is only false if both propositions are false.

The truth table for the disjunction “p or q” is:

p q p _ q

T T T

T F T

F T T

F F F

p _ q is true if p or q or both are true.

p _ q is only false if both p or q are false.

If P and Q are the truth sets for propositions p and qrespectively, then the truth set for p _ q is P [Q,

the region where p or q or both are true.

U

P Q

U

P Q

P Q[is shaded

5 is a prime number.

A triangle has sides.five



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EXCLUSIVE DISJUNCTION

The exclusive disjunction is true when only one of the propositions is true.

The exclusive disjunction of p and q is written p Y q.

We can describe p Y q as “p or q, but not both”, or “exactly one of p and q”.

So, if p: Sally ate cereal for breakfast

and q: Sally ate toast for breakfast

then p Y q: Sally ate cereal or toast, but not both, for breakfast.

The truth table for the exclusive disjunction p Y q is:

p q p Y q

T T F

T F T

F T T

F F F

p Y q is true if exactly one of p and q is true.

p Y q is false if p and q are both true or both false.

If P and Q are the truth sets for propositions p and qrespectively, then the truth set for pY q is the region

shaded, where exactly one of p and q is true.

EXERCISE 17B.2

1 Write the disjunction p _ q for the following pairs of propositions:

a p: Tim owns a bicycle, q: Tim owns a scooter.

b p: x is a multiple of 2, q: x is a multiple of 5.

c p: Dana studies Physics, q: Dana studies Chemistry.

2 For the following propositions, determine whether p _ q is true or false:

a p: 24 is a multiple of 4, q: 24 is a multiple of 6.

b p: There are 100o in a right angle, q: There are 180o on a straight line.

c p: ¡8 > ¡5, q: 5 < 0

d p: The mean of 5 and 9 is 7, q: The mean of 8 and 14 is 10.

3 Write the exclusive disjunction p Y q for the following pairs of propositions:

a p: Meryn will visit Japan next year, q: Meryn will visit Singapore next year.

b p: Ann will invite Kate to her party, q: Ann will invite Tracy to her party.

c p: x is a factor of 56, q: x is a factor of 40.

4 For the following pairs of propositions a and b, determine whether the exclusive

disjunction a Y b is true or false:

a a: 23 is a prime number, b: 29 is a prime number.

b a: 15 is even, b: 15 is a multiple of 3.

U

P Q



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LOGIC (Chapter 17) 537

c a: 4:5 2 Z , b: ¡8 2 N

d a: 23 £ 24 = 27, b: (28)6 = 242

5 Consider r: Kelly is a good driver, and s: Kelly has a good car.

Write in symbolic form:

a Kelly is not a good driver

b Kelly is a good driver and has a good car

c Kelly does not have a good car and is not a good driver

d Kelly has a good car or Kelly is a good driver.

6 Consider x: Sergio would like to go swimming tomorrow, and

y: Sergio would like to go bowling tomorrow.

Write in symbolic form:

a Sergio would not like to go swimming tomorrow

b Sergio would like to go swimming and bowling tomorrow

c Sergio would like to go swimming or bowling tomorrow

d Sergio would not like to go both swimming and bowling tomorrow

e Sergio would like either to go swimming or go bowling tomorrow, but not both.

7 For each of the following, define appropriate propositions and then write in symbolic

form:

a Phillip likes icecream and jelly.

b Phillip likes icecream or Phillip does not like jelly.

c x is both greater than 10 and a prime number.

d Tuan can go to the mountains or the beach, but not both.

e The computer is not on.

f Angela does not have a watch but does have a mobile phone.

g Maya studied one of Spanish or French.

h I can hear thunder or an aeroplane.

8 If p _ q is true and p Y q is false, determine the truth values of:

a p b q

9 For U = fx j 1 6 x 6 20, x 2 Z g, consider the propositions

p: x is a multiple of 3, and q: x is an odd number.

a Illustrate the truth sets for p and q on a Venn diagram.

b Use your Venn diagram to find the truth set for:

i :q ii p _ q iii p ^ q iv p Y q

10 a For U = fx j 1 6 x 6 12, x 2 Z g, use a Venn diagram to illustrate the truth sets

for p: x is prime, and q: x is a factor of 12.

b Write down the meaning of:

i p ^ q ii p _ q iii p Y q

c Use your Venn diagram to find the truth sets for:

i p ^ q ii p _ q iii p Y q


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11 Read the description of Ed’s day:

“Ed slept in, then had pancakes for breakfast. He went to the gym in the morning, then

ate a sandwich for lunch. He played golf in the afternoon, and had steak for dinner.”

Consider the following propositions:

p: Ed got out of bed early

r: Ed ate steak for lunch

t: Ed ate fish for dinner

v: Ed went to the movies

q: Ed ate pancakes for breakfast

s: Ed ate steak for dinner

u: Ed went to the gym

w: Ed played golf.

Determine whether the following are true or false:

a p b s c q ^ u d p _w e r _ s f r ^ s g r Y s h t _ v

Let P be the truth set of proposition p and Q be the truth set of proposition q.

Use mathematical logic to express the following shaded regions in terms of p and q:

a b c

a The shaded region is P [Q, which is the region in P or Q or both.

So, p or q or both are true, which is p _ q.

b The shaded region is Q0, which is the region not in Q.

So, q is not true, which is :q.

c The shaded region is P 0 \Q, which is the region in Q but not in P .

So, p is not true and q is true, which is :p ^ q.

12 Let P be the truth set of proposition p and Q be the truth set of proposition q. Use

mathematical logic to express the following shaded regions in terms of p and q:

a b c

13 For each of the following propositions:

i write down the corresponding set notation ii illustrate on a Venn diagram.

a p _ q b :p _ q c p Y q d :p ^ :q14

a b c


P Q

U

P Q

UU

P Q

P Q

U

P Q

U

P Q

U

A B

U

A B

U

A B

U

Consider a: The captain is male and b: The captain is old.

Let A be the truth set of a and B be the truth set of b.Interpret each of the following Venn diagrams:


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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_17\538IB_STSL-2_17.cdr Monday, 15 February 2010 9:49:18 AM PETER

p qNegation

:pConjunction

p ^ qDisjunction

p _ q

Exclusive

disjunction

p Y q

T T F T T F

T F F F T T

F T T F T T

F F T F F F

We can use these rules to construct truth tables for more complicated propositions.

Construct a truth table for p _ :q.

To construct a truth table for p_:q, we use the negation rule on q to find :q, then

use the disjunction rule on p and :q to find p _ :q.

We start by listing all the possible

combinations for p and q:

p q :q p _ :qT T

T F

F T

F F

We then use the negation rule on the

q column to find :q:

p q :q p _ :qT T F

T F T

F T F

F F T

Finally, we use the disjunction rule on

the p and :q columns to find p _ :q:

p q :q p _ :qT T F T

T F T T

F T F F

F F T T

TAUTOLOGY AND LOGICAL CONTRADICTION

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

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

are false.


C TRUTH TABLES AND

LOGICAL EQUIVALENCE

The truth tables for negation, conjunction, disjunction, and exclusive disjunction can be

summarised in one table.



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For any proposition p,

either p is true or :p

is true. So, p _ :p is

always true.

Show that p _ :p is a tautology.

The truth table is: p :p p _ :pT F T

F T T

All the values in the p _ :p column are true, so

p _ :p is a tautology.

LOGICAL EQUIVALENCE

Two propositions are logically equivalent if they have the same truth table column.

Show that :(p ^ q) and :p _ :q are logically equivalent.

The truth table for :(p ^ q) is:

p q p ^ q :(p ^ q)

T T T F

T F F T

F T F T

F F F T

The truth table for :p _ :q is:

p q :p :q :p _ :qT T F F F

T F F T T

F T T F T

F F T T T

As the truth table columns for :(p ^ q) and :p _ :q are identical,

:(p ^ q) and :p _ :q are logically equivalent.

So, :(p ^ q) = :p _ :q.

EXERCISE 17C.1

1 Construct a truth table for the following propositions:

a :p ^ q b :(p Y q) c :p _ :q d p _ p

2 For the following propositions:

i construct a truth table for the proposition

ii determine whether the proposition is a tautology, a logical contradiction, or

neither.

a :p ^ :q b (p _ q) _ :p c p ^ (p Y q) d (p ^ q) ^ (p Y q)

3 a Explain why p ^ :p is always false.

b Use a truth table to show that p ^ :p is a logical contradiction.





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4 Use truth tables to establish the following logical equivalences:

5 a Construct a truth table for (:p ^ q) _ (p ^ :q).b Use the truth table summary at the start of this section to identify a proposition

logically equivalent to (:p ^ q) _ (p ^ :q).

6 a Consider the propositions p: I like apples and q: I like bananas.

Write the meaning of:

i p _ q ii :(p _ q) iii :p iv :p ^ :qb Use truth tables to show that :(p _ q) and :p ^ :q are logically equivalent.

7 a Complete the truth table below:

p q p Y q q ^ (p Y q) (p Y q) _ p

T T

T F

F T

F F

b Consider the propositions p: ¡3 6 x 6 7 and q: x > 2.

Find the values of x which make the following propositions true:

i p Y q ii q ^ (p Y q) iii (p Y q) _ p

8 Explain why:

a any two tautologies are logically equivalent

b any two logical contradictions are logically equivalent.

9 What can be said about:

a the negation of a logical contradiction

b the negation of a tautology

c the disjunction of a tautology and any other statement?

TRUTH TABLES FOR THREE PROPOSITIONS

p q rT T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

When three propositions are under consideration, we usually

denote them p, q and r.

The possible combinations of the truth values for p, q and r are

listed systematically in the table alongside.


a :(:p) = p b p ^ p = p c p _ (:p ^ q) = p _ q

d :(p Y q) = p Y :q e :(q _ :p) = :q ^ (p _ q) f :p Y (p _ q) = p _ :q


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Construct a truth table for

the compound proposition

(p _ q) ^ r.

To find (p _ q) ^ r, we must first find p _ q.

We then find the conjunction of p _ q and r.

p q r p _ q (p _ q) ^ rT T T T T

T T F T F

T F T T T

T F F T F

F T T T T

F T F T F

F F T F F

F F F F F



EXERCISE 17C.2

1 Construct truth tables for these compound statements:

a :p _ (q ^ r) b (p _ :q) ^ r c (p _ q) _ (p ^ :r)2 Determine whether the following propositions are tautologies, logical contradictions, or

neither:

a (p _ q) _ :(r ^ p) b (p Y r) ^ :q c (q ^ r) ^ :(p _ q)

3 a Consider the propositions p: Jake owns a phone

q: Jake owns a TV

r: Jake owns a laptop.

Write down the meaning of:

i p ^ q ii (p ^ q) ^ r iii q ^ r iv p ^ (q ^ r)

b Use truth tables to show that (p ^ q) ^ r = p ^ (q ^ r).

4 Use truth tables to show that (p _ q) _ r and p _ (q _ r) are logically equivalent.

5 a Consider the propositions p: Mary will study Mathematics next year

q: Mary will study French next year

r: Mary will study German next year.

Write down the meaning of:

i q _ r ii p ^ (q _ r) iii p ^ q iv p ^ r v (p ^ q) _ (p ^ r)

b Use truth tables to show that p ^ (q _ r) = (p ^ q) _ (p ^ r).

6 a Use truth tables to show that p _ (q ^ r) = (p _ q) ^ (p _ r).

b Consider the Venn diagram alongside, where

P , Q, and R are the truth sets of p, q, and

r respectively.

On separate Venn diagrams, shade the truth

set for:

i p _ (q ^ r)

ii (p _ q) ^ (p _ r)

Comment on your results.

P Q

RU


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IMPLICATION

If two propositions can be linked with “If ...., then ....”, then we have an implication.

The implicative statement “if p then q” is written p ) q and reads “p implies q”.

p is called the antecedent and q is called the consequent.

For example, if p: Kato has a TV set, and q: Kato can watch TV,

then we have p ) q: If Kato has a TV set, then Kato can watch TV.

THE TRUTH TABLE FOR IMPLICATION

Consider p: It will rain on Saturday, and q: The Falcons will win.

The implicative statement is p ) q: If it rains on Saturday, then the Falcons will win.

To establish the truth table for p ) q, we will consider each of the possible combinations

for p and q in turn:

So, the truth table for p ) q is:

EQUIVALENCE

If two propositions are linked with “.... if and only if ....”, then we have an equivalence.

The equivalence “p if and only if q” is written p , q.

IMPLICATION AND EQUIVALENCED

p q Scenario p ) q

T T It rains on Saturday, and the Falcons win.

This is consistent with the implicative statement.

T

T F It rains on Saturday, but the Falcons do not win.

This is inconsistent with the implicative statement.

F

F T It does not rain on Saturday, and the Falcons win.

This is consistent with the implicative statement, as no claim

has been made regarding the outcome if it does not rain.

T

F F It does not rain on Saturday, and the Falcons do not win.

Again, this is consistent with the implicative statement as no

claim has been made regarding the outcome if it does not rain.

T

p q p ) q

T T T

T F F

F T T

F F T

p ) q is only false if

p is true but q is false.



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p , q is logically equivalent to the conjunction of the implications p ) q and q ) p.

Consider p: I will pass the exam, and q: The exam is easy.

We have p ) q: If I pass the exam, then the exam is easy.

q ) p: If the exam is easy, then I will pass it.

p , q: I will pass the exam if and only if the exam is easy.

THE TRUTH TABLE FOR EQUIVALENCE

We can find the truth table for p , q by constructing the truth table of its logical equivalent

(p ) q) ^ (q ) p):

p q p ) q q ) p (p ) q) ^ (q ) p)

T T T T T

T F F T F

F T T F F

F F T T T

So, the truth table for equivalence p , q is:

p q p , q

T T T

T F F

F T F

F F T

EXERCISE 17D

1 In the following implicative statements, state the antecedent and the consequent.

a If I miss the bus, then I will walk to school.

b If the temperature is low enough, then the lake will freeze.

c If x > 20, then x > 10.

d If you jump all 8 hurdles, then you may win the race.

2 For the following propositions, write down the implicative statement p ) q:

a p: The sun is shining, q: I will go swimming

b p: x is a multiple of 6, q: x is even

c p: There are eggs in the fridge, q: Jan will bake a cake.

3 For the following propositions p and q:

i write down the equivalence p , q

ii state whether the equivalence is true or false.

a p: Rome is the capital of Italy, q: Paris is the capital of France

b p: 2x+ 3 = 10 is an expression, q: 2x+ 3 is an expression

c p: Cows have nine legs, q: Horses have five heads.

The equivalence

p , q is true when p

and q have the same

truth value.



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4 Consider the propositions p: It is raining and q: There are puddles forming.

Write the following statements in symbols:

a If it is raining then puddles are forming.

b If puddles are forming then it is raining.

c Puddles are not forming.

d It is not raining.

e If it is not raining, then puddles are not forming.

f If it is raining, then puddles are not forming.

g If there are no puddles, then it is raining.

h It is raining if and only if there are puddles forming.

5 Construct truth tables for:

a p ) :q b :q ) :p c (p ^ q) ) p d q ^ (p ) q)

e p , :q f (p , q) ^ :p g p ) (p ^ :q) h (p ) q) ) :p6 By examining truth tables, show that:

a p Y q = :(p , q) b :p ) q = p _ q

c q ) (p Y q) = :(p ^ q) d p , q = (p ^ q) _ (:p ^ :q)7 Which of these forms are logically equivalent to the negation of q ) p?

8 Determine whether the following are logical contradictions, tautologies, or neither:

a p ) (:p ^ q) b p ^ q ) p _ q c (p ) :q) _ (:p ) q)

THE CONVERSE

The converse of the statement p ) q is the statement q ) p.

The converse has truth table: p q q ) p

T T T

T F T

F T F

F F T

For p: the triangle is isosceles, and q: two angles of the triangle are equal,

state p ) q and its converse q ) p.

p ) q: If the triangle is isosceles, then two of its angles are equal.

q ) p: If two angles of the triangle are equal, then the triangle is isosceles.

CONVERSE, INVERSE AND

CONTRAPOSITIVE

E


A p ) q B :q ) p C q ) :p D :(:p ) :q)


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p q :p :q :p ) :qT T F F T

T F F T T

F T T F F

F F T T T

p q :q :p :q ) :pT T F F T

T F T F F

F T F T T

F F T T T

Implication

p ) qIf Sam is in the library, then Sam is reading.

Converse

q ) pIf Sam is reading, then Sam is in the library.

Inverse

:p ) :q If Sam is not in the library, then Sam is not reading.

Contrapositive

:q ) :p If Sam is not reading, then Sam is not in the library.

logically

equivalent


THE INVERSE

The inverse statement of p ) q is the statement :p ) :q:

The inverse has truth table:

This is the same truth table as q ) p:

So, the converse and inverse of an implication are logically equivalent.

THE CONTRAPOSITIVE

The contrapositive of the statement p ) q is the statement :q ) :p.

The contrapositive has truth table:

The truth table for :q ) :p is the same as that for p ) q.

So, the implication and its contrapositive are logically equivalent.

For example, consider p: Sam is in the library and q: Sam is reading.

The implication and the converse are not logically equivalent since, for example, the

implication allows for the possibility that Sam is reading in the classroom, but the converse

does not.

EXERCISE 17E

1 Write the converse and inverse for:

a If Nicole is wearing a jumper, then she is warm.

b If two triangles are similar, then they are equiangular.

c If 2x2 = 12, then x = §p6:


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d If Alex is in the playground, then he is having fun.

e If a triangle is equilateral, then its three sides are equal in length.

Write down the contrapositive of: “All teachers drive blue cars”.

This statement is the same as “if a person is a teacher, then he or she drives a blue car”.

This has the form p ) q with p: A person is a teacher and q: A person drives a

blue car.

The contrapositive :q ) :p is “If a person does not drive a blue car, then the person

is not a teacher.”

2 Write down the contrapositives of these statements:

a All rose bushes have thorns.

b All umpires make correct decisions all the time.

c All good soccer players have good kicking skills.

d Liquids always take the shape of the container in which they are placed.

e If a person is fair and clever then the person is a doctor.

3 a State the contrapositive of: “All high school students study Mathematics.”

b Suppose the statement in a is true. What, if anything, can be deduced about:

i Keong, who is a high school student

ii Tamra, who does not study Mathematics

iii Eli, who studies English and Mathematics?

4 Write down the contrapositive of:

a x is divisible by 3 ) x2 is divisible by 9

b x is a number ending in 2 ) x is even

c PQRS is a rectangle ) PQ kSR and PS kQR

d KLM is an equilateral triangle ) KbML measures 60o.

5 Consider p: A house has at least 3 windows and q: A house has a chimney.

The implication is p ) q: If a house has at least 3 windows, then it has a chimney.

a For this implication, write down the:

i converse ii inverse iii contrapositive.

b Determine the truth values for the implication, converse, inverse, and contrapositive

for each of these houses:

i ii iii



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WE

U

VALID ARGUMENTS (EXTENSION)F

If George is at the beach, then he is getting sunburnt.

George is at the beach.

George is getting sunburnt.

premise

conclusion

p ) q

p

q

premise

conclusion


6 W represents all weak students and E represents all

Year 11 students.

a Copy and complete:

i No weak students are ....

ii No Year 11 students are ....

b Copy and complete:

i If x 2 W then ....

ii If x 2 E then ....

c

An argument is made up of a set of propositions, called the premise, that leads to a

conclusion.

An argument is usually indicated by the words ‘therefore’ or ‘hence’.

A simple example of an argument is:

If George is at the beach, then he is getting sunburnt.

George is at the beach.

Therefore, George is getting sunburnt.

We set out arguments by separating the premise and the conclusion with a horizontal line.

We can test whether the logic applied in our argument is valid by expressing the argument in

terms of propositions.

If we have p: George is at the beach,

and q: George is getting sunburnt,

then the argument becomes:

So, from the propositions p ) q and p, we are implying the conclusion q. We can write

this argument in logical form as (p ) q) ^ p ) q.

To determine whether this

argument is valid, we

construct a truth table for this

proposition, and see whether

it is a tautology.

Since we have a tautology, we can say that our argument is valid. This means that the

conclusion we have made follows logically from the premise.

What is the relationship between the implications in ?b

p q p ) q (p ) q) ^ p (p ) q) ^ p ) q

T T T T T

T F F F T

F T T F T

F F T F T


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IMPORTANT: Proposition q is clearly false. However, this does not affect the validity of

the argument. Logic is not concerned with whether the premise is true or

false, but rather with what can be validly concluded from the premise.

EXERCISE 17F.1

1 Consider the following argument:

Lucy will have to work if and only if Paul is sick.

Paul is not sick.

Therefore, Lucy will not have to work today.

a Write down the premise and conclusion of this argument in terms of the propositions

p: Lucy will have to work today and q: Paul is sick.

b Write the argument in logical form.

c Construct a truth table to show that the argument is valid.

The validity of anargument is not related tothe actual truth values ofthe propositions within it.

Determine the validity of the following argument:

If a triangle has three sides, then 2 + 4 = 7.

2 + 4 = 7Hence, a triangle has three sides.

We have p: A triangle has three sides and q: 2 + 4 = 7

The argument is: p ) q

q

p

premise

conclusion

We can write this in logical form as (p ) q) ^ q ) p.

p q p ) q (p ) q) ^ q (p ) q) ^ q ) p

T T T T T

T F F F T

F T T T F

F F T F T

Since we do not have a tautology, the argument is not valid.



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2 a Write the following arguments in logical form:

i p ) q

:q:p

ii p _ q

:pq

iii p _ q

p

iv p ) q

:p:q

v p ) q

q ) p

p

b Construct truth tables for each part in a. Which of the arguments are valid?

3 Determine the validity of the following arguments written in logical form:

a (p ^ q) ) p b (p ) q) ^ :q ) p

c (p ) q) ^ (q ) p) ) (p , q) d (p ^ :q) ) (:p _ q)

4 Use p: x is prime and q: x is odd to show that the following are valid arguments:

a If x is prime, then x is odd.

x is prime or odd.

Hence, x is odd.

b x is prime or odd, but not both.

x is not odd.

Therefore, x is prime.

5 Consider the following argument: Don has visited Australia or New Zealand.

Don has visited New Zealand.

Therefore, Don has not visited Australia.

a Use a truth table to show that this argument is invalid.

b Describe the scenario which demonstrates that the argument is invalid.

6 Determine the validity of the following arguments:

a Tan went to the movies or the theatre last night, but not both.

Tan did not go to the movies.

Therefore, Tan went to the theatre.

b If x is a multiple of 4, then x is even.

Hence, if x is even, then x is a multiple of 4.

c London is in China if and only if 20 is a multiple of 5.

20 is a multiple of 5.

Therefore, London is in China.

d x is a factor of 30 or 50.

Hence, x is a factor of 50.

e If the sequence is not geometric, then the sequence is arithmetic.

Therefore, the sequence is arithmetic or geometric.

f All students like chips.

Melanie likes chips.

Hence, Melanie is a student.


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SYLLOGISMSINVESTIGATION

A syllogism is an argument consisting of three lines. The third line is

supposed to be the logical conclusion from the first two lines.

Example 1:

If I had wings like a seagull I could fly.

I have wings like a seagull.

Therefore, I can fly.

Example 2:

If I had wings like a seagull I could fly.

I can fly.

Therefore, I have wings like a seagull.

The arguments in these examples can be written as:

Example 1: p ) q

p

q

Example 2: p ) q

q

p

What to do:

1 Use truth tables to show that the first example is a valid argument, and the second

is invalid.

2 Consider this syllogism: All cows have four legs.

Wendy is not a cow.

Hence, Wendy does not have four legs.

We see it is comprised of two propositions, p: x is a cow, and

q: x has four legs.

Write the argument in logical form and show that it is invalid.

3 Test the validity of the following syllogisms:

a All prime numbers greater than two are odd. 15 is odd.

Hence, 15 is a prime number.

b All mathematicians are clever. Jules is not clever.

Therefore, Jules is not a mathematician.

c All rabbits eat grass. Peter is a rabbit.

Therefore, Peter eats grass.

4 Give the third line of the following syllogisms to reach a correct conclusion.

a All cats have fur.

Jason is a cat.

Therefore, ......

b Students who waste

time fail.

Takuma wastes time.

Hence, ......

c All emus cannot fly.

Fred can fly.

Therefore, ......



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ARGUMENTS WITH THREE PROPOSITIONS

Determine the validity of the following argument:

If x is a natural number, then x is an integer.

If x is an integer, then x is rational.

Therefore, if x is a natural number, then x is rational.

We have p: x is a natural number, q: x is an integer, and r: x is rational.

The argument is written as p ) q

q ) r

p ) r

We can write this in logical form as (p ) q) ^ (q ) r) ) (p ) r).

The logical form of the argument is a tautology, so the argument is valid.

EXERCISE 17F.2

1 Consider the propositions p: It is sunny, q: I am warm, and r: I feel happy.

a (p ^ q) ) r b p ^ :q ) :r c q ^ r ) p

2 Which, if any, of the following arguments are valid?

3 a Show that the argument p ) q

q ) r

p ) :r

is invalid.

b What truth values of p, q and r lead to an invalid argument?


p q r p ) q q ) r (p ) q) ^ (q ) r) p ) r (p ) q) ^ (q ) r)

) (p ) r)

T T T T T T T T

T T F T F F F T

T F T F T F T T

T F F F T F F T

F T T T T T T T

F T F T F F T T

F F T T T T T T

F F F T T T T T


A p ) q

q ) r

(q ^ r)

B (p ^ q) _ r

p _ r

C (p ^ q) ) r

p

r

Write the following arguments in words.


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REVIEW SET 17A


4 If I do not like a subject, I do not work hard. If I do not work hard I fail. I passed,

therefore I must like the subject.

a Identify the propositions p, q and r.

b Write the above argument in logical form.

c Is the conclusion a result of valid reasoning?

5 Determine the validity of this argument:

If Jeremy is on the basketball team, then he is tall and fast.

Jeremy is tall and he is not on the basketball team.

Therefore, Jeremy is not fast.

1 Which of the following are propositions? If they are propositions, state whether they

are true, false or indeterminate.

a Sheep have four legs. b Do giraffes have four legs?

c Alicia is good at Mathematics. d I think my favourite team will win.

e Vicki is very clever. f There are 7 days in a week.

g Put your shoes on. h All cows are brown.

i a2 + b2 = c2

j The opposite sides of a parallelogram are equal.

2 Consider the propositions p: x is an even number, and q: x is divisible by 3.

Write the following in words:

a :p b p _ q c p Y q d p ) q

e :p ^ q f :p Y q g p ) :q h :p ) :q3 Consider the propositions p: x is a prime number, and q: x is a multiple of 7.

Write the following in symbolic language:

a If x is a prime number then x is a multiple of 7.

b x is not a prime number.

c x is a multiple of 7 and not a prime number.

d x is either a prime number or a multiple of 7 but not both.

e x is neither a prime number nor a multiple of 7.

In each case, write down a number that satisfies the statement.

4 Write the implication p ) q, the inverse, converse and contrapositive of the

following propositions in both words and symbols.

a p: I love swimming.

q: I live near the sea.

b p: I like food.

q: I eat a lot.

5 Represent the truth sets of the following on Venn diagrams:

a p Y q b :(p _ q) c :p ^ q

d :p e :p _ q f :(p ^ q ^ r)


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REVIEW SET 17B


6 For the propositions p: x is a factor of 12, and q: x is an odd number < 10,

list the truth sets of:

a p b q c p ^ q d p _ q

7 Use truth tables to determine the validity of the following arguments:

a p ) q

:p:q

b p _ q

:q:p

c p ) q

q ) r

r _ q

1 Consider the propositions p: x is a multiple of 4, 18 < x < 30

q: x is a factor of 24,

and r: x is an even number, 18 < x < 30.

a List the truth sets of p, q, r, and p ^ q ^ r.

b List the elements of: i p ^ q ii q ^ r iii p ^ r

2 Find negations for the following:

a Eddy is good at football.

b The maths class includes more than 10 boys.

c The writing is illegible. d Ali owns a new car.

3 Write the following statements as implications:

a All birds have two legs. b Snakes are not mammals.

c No rectangle has five sides. d This equation has no real solutions.

4 ‘Positive’ and ‘negative’ are defined as follows:

x is positive , x > 0 x is negative , x < 0

a Is zero positive or negative?

b What is the negation of ‘x is negative’ when x 2 frational numbersg?

5 Let P , Q and R be the truth sets of propositions p, q and r respectively.

Write the following as compound propositions in terms of p, q and r:

a b c

6 Which of the following pairs are logically equivalent?

a p ) q and :q ) :p b :(p ^ q) and :p _ :qc p , q and (p ^ q) ^ :q d :p ) :q and q ) p

U

P

QU

P

QU

P

Q

R


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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_17\554IB_STSL-2_17.cdr Monday, 15 February 2010 9:57:50 AM PETER

REVIEW SET 17C


7 Express the following in logical form. Determine whether or not the argument is

valid.

a If the sun is shining I will wear my shorts. The sun is shining. Therefore, I

wear shorts.

b All teachers work hard. Marty is not a teacher. Therefore Marty does not work

hard.

1 Find the negation of:

a x 6 3 for x 2 Z

b x is a comb, for x 2 fbrush, comb, hairclip, bobby pingc x is a tall woman for x 2 fwomeng

2 For U = fx j 1 6 x 6 20g, consider the propositions

p: x is an even number, and q: x is a square number.

a Illustrate the truth sets for p and q on a Venn diagram.

b Use your Venn diagram to find the truth set for:

i p ^ q ii :p _ q iii :(p Y q)

3 Write down, in words, the inverse, converse and contrapositive for the implication:

“The diagonals of a rhombus are equal.”

4 Consider the propositions p: cakes are sweet, and q: cakes are full of sultanas.

Write each of the following using logic symbols:

a If cakes are not sweet they are not full of sultanas.

b If cakes are not sweet they are full of sultanas.

c Cakes are full of sultanas and they are not sweet.

d Cakes are not sweet or they are full of sultanas.

5 Consider the propositions:

p: The plane leaves from gate 5. q: The plane leaves from gate 2.

r: The plane does not leave this morning.

a Write the following logic statement in words: p ) (:r ^ :q)b Write in symbols: The plane leaves this morning if and only if it leaves from

gate 2 or from gate 5.

6 Construct truth tables for the following and state whether the statements are

tautologies, logical contradictions or neither:

a (p ) q) ^ q ) p b (p ^ q) ^ :(p _ q) c :p , q

d (p _ :q) ) q e (:p _ q) ) r f p ^ q ) q


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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_17\555IB_STSL-2_17.cdr Tuesday, 16 March 2010 11:52:45 AM PETER


7 Express the following in logical form. Determine whether or not the argument is

valid.

a If Fred is a dog he has fur. If Fred has fur he has a cold nose.

Fred is a dog. Hence, Fred has a cold nose.

b If Viv is a judge, she wears a robe or a wig.

Viv does not wear a wig, nor is she a judge.

Therefore, Viv does not wear a robe.


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Y:\HAESE\IB_STSL-2ed\IB_STSL-2ed_17\556IB_STSL-2_17.cdr Friday, 12 February 2010 4:10:19 PM PETER

17 - smaris.edu.ecsmaris.edu.ec/wp-content/uploads/2013/08/logic.pdf · chapter 17 logic contents:...

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