Mathematics LE

Key Unit competence: Use Mathematical logic as a tool of reasoning and decision making in daily life

2.0 INTRODUCTORY ACTIVITY 2

A) Discuss the meaning of propositional statement

B) In the following sentences which of them are propositions and which are not propositions?

1. Don’t eat the daisies!

2. My dog is called Didi.

3. Do you enjoy reading novels?

4. The jokes are great.

5. Mozart composed classical music.

6. The camera is not a Kodak.

7. This statement is false.

8. Use the quadratic formula on that one.

2.1 Simple statement and compound statements

ACTIVITY 2.1

From the following expressions, give your answer by true or false

1. Every integer larger than 1 is positive

2. Kampala is in Rwanda

3. How old are you?

4. Every liquid is water.

5. Write down the names of Rwandan president.

6. 1− x² = 0 .

7. Rwanda is an African country or Rwanda is a member of Commonwealth.

Content summary

A sentence which is either true or false but not both simultaneously is named statement or proposition. In the context of logic, a proposition or a statement is the sentence in the grammatical sense conveying a situation which is neither imperative, interrogative nor exclamatory.

The expressions 1, 2, 4 and 7 are statements: the 2nd and 4th are false while 1st and 7th are true.

• • The expression “How old are you?” is not a proposition since you cannot reply by true nor false (grammatically this sentence is interrogative).
•  The equality “1− x2 = 0 ” is not a proposition because for some values of x the equality is true, whereas for others it is false.
•  The expression “Write down the names of Rwandan president” is not a proposition as the answer will be given by neither true nor false. It is a command.

A statement that cannot be broken into two or more sentences is called simple statement. Combining two or more simple statements we form a compound statement.

Example: In activity 7.1.1, the 1st, 2nd and 4th expressions are simple statements while the 7th is a compound statement.

In this unit, statements will be denoted by small letters such as p, q, r, ...

The logical statements are required to have a definite truth-value, or, to be either true or false, but never both, and to always have the same truth value.

The two truth values of proposition are true and false and are denoted by the symbols T and F respectively.

 Occasionally, they are also denoted by the symbols 1 and 0 respectively.

APPLICATION ACTIVITY 2.1

1) Find out which of the following sentences are statements and which are not. Justify your answer.

a) Uganda is a member of East African Community.

b) The sun is shining.

c) Come to class!

d) The sum of two prime numbers is even.

e) It is not true that China is in Europe.

f) May God bless you!

2) Write down the truth value (T or F) of the following statements

a) Paris is in Italy.

b) 13 is a prime number.

c) Kigeri IV Rwabugiri was the king of the Kingdom of Rwanda

d) Lesotho is a state of South Africa.

2.2 Truth values and truth tables

ACTIVITY 2.2

Are these sentences proposition? If yes, give their truth values

a) Uganda is a member of East African Community.

b) The sun shines.

c) Paris is in England.

d) Come to class!

e) The sum of two prime numbers is even.

f) It is not true that Uganda is in Europe.

Content summary

The way we will define compound statements is to list all the possible combinations of the truth-values (abbreviated T and F or 1and 0) of the simple statements (that are being combined into a compound statement) in a table, called a truth table. The name of each statement is at the top of a column of the table.

If the compound statement contains n distinct simple statements, we will consider 2n possible combinations of truth values in order to obtain the truth table.

Examples

1) One proposition p has two truth values ( 2¹ = 2 possible combinations), the truth table is

2) Suppose we are given two mathematical statements, named P and Q, new mathematical statements that incorporate P and Q are called compound statements. Their truth values will be determined solely by the truth-values of P and of Q.

For two propositions p and q, we have 2² = 4 possible combinations of truth values, the truth table is:

Any proposition can be represented by a truth table

•  It shows truth values for all combinations of its constituent variables

Example:

Proposition involving 2 variables p and q. Let us take the true statement “p and q” where p: I am at home, and q: It is raining. One combination of r = p ∧ q is I am at home and it is raining. Others are the following: I am at home and it is not raining (F), I am not at home and it is raining (F) and I am not at home and it is not raining (F).

A proposition can involve any number of variables; each row corresponds to a possible combination of variables. With n variables the truth table has:

n+1 column (one for each of the n variables and one for the compound expression);

It has also 2ⁿ rows plus a header.

The following lesson will develop how to make a compound statement using different connectives.

APPLICATION ACTIVITY 2.2

Write down the truth table for

1) Three propositions p, q and r

2) Four propositions p, q, r and s

2.3 Logical connectives

2.3.1 Negation

ACTIVITY 2.3.1

1) Given statements P: I am strong and Q: I can jump, try to make a compound statement formed by P and Q in different ways. What are the different connecting words that can be used?

2) Let p, q, … are the given propositions, put these propositions in negative form

1. Jack is running.

2. Ronald does not smile.

3. She isn’t a foot ball player.

4.    -3 is a natural number.

5. Mathematics is needed in languages Education option.

Content summary

The negation of a statement P is made by introducing the word “not” denoted by prefixing the statement P. It has opposite truth value from the statement. It is denoted by  

From this definition, it follows that the negation of a true statement is false while the negation of false statement is true; simply If P is true, then ¬P is false and if P is false, then ¬P true.

Example

1) Let P : “Kamana is a student”, then¬P : “Kamana is not a student”.

2)P: The earth is round, ¬P : The earth is not round.

Let P be a proposition. Construct the truth table of ¬P

Solution

APPLICATION ACTIVITY 2.3.1

1. Write the negation of each of the following statements:

     a) Today is raining.

     b) The sky is blue

     c) My native country is Rwanda.

     d) Bony is smart and healthy.

2. Complete the following truth table

2.3.2 Conjunction

ACTIVITY 2.3.2

Given two propositions which are true;

p: I am at school, q: it is raining, Discuss the truth value of the compound propositions:

a) “I am at school and it is raining”

b) “I am not at school and it is raining”,

c)“I am not at school and it is not raining”.

Content summary

If two simple statements p and q are connected by the word “and”, then the resulting compound statement p and q is called a conjunction of p and q and is written in symbolic form p ∧ q . It has the truth value true whenever both p and q have the truth value true; otherwise it has the truth value false.

Examples

Let p be “It is raining today” and q be “there are fifteen chairs in this class room” assuming that p and q are true statements, construct simple sentences which describe each of the following statements and construct the truth table.

a) p ∧ q

Solution

a) From the given two simple statements, one of the resulting compound statements is

(i) “It is raining today and there are fifteen chairs in this class room” which is True.

(ii) “It is not raining and there are not fifteen chairs in this classroom: Which is False.

You can give the remaining two sentences and the truth table of related compound statements is as follows:

2) Let p and q be propositions. Construct the truth table for

a) ¬p ∧ q

b) (¬p ∧ q) ∧¬p

Solution

APPLICATION ACTIVITY 2.3.2

1) If p stands for the statement “ It is cold” and q stands for the statement

“ It is raining”, then what does ¬q ∧¬q stands for? Construct its truth table

2) Let p and q be two propositions. Construct the truth table of

a) p ∧ q

b) ¬p ∧ q

c) p ∧¬q

d) ¬( p ∧ q)

e) ¬q ∧(¬p ∧ q)

3) Determine the truth value of each of the following statements

1. Paris is in France and it is a Capital city

2. 4 + 4 = 9 and 5 + 8 =11

3. Paris is in England and 3+ 4 = 7

4. Kigali is the Capital city of Burundi and1+1 = 2

5. Alphabets are the basic of any languages and the digits is not the basic in counting

6. m² is the unit of area and kg being one of the units of weight.

2.3.3 Disjunction

ACTIVITY 2.3.3

1)There were boys and girls in the classroom.

a) If they ask you to choose 2 boys and two girls. How many students will you choose?

b) If they ask you to select two girls or two boys how many students will you select?

2) Given the true proposition p: I am at home, q: it is raining. What is the truth value of

a) “I am at home or it is raining”.

b) “I am not at home or it is not raining”.

c) “I am at home or it is not raining”.

Content summary

If two simple statements p and q are connected by the word “or”, then the resulting compound statement “p or q” is called a disjunction of p and q and it is written in symbolic form by p ∨ q .

It has the truth value false only when p and q have truth value false, otherwise it has true as a the truth value.

Examples

1) Let p be “Paris is in France” and q be “London is in England”. Construct simple verbal sentences which describe different related compound statements and construct the truth table. of p ∨ q .

Solution

From the given two simple statements, the resulting compound statement is

(i) “Paris is in France or London is in England” (True)

(ii) Paris is in France or London is not in England (Tue)

(iii) Paris is not in France or London is in England (True)

(iii) Paris is not in France or London is not in England (False).

The Truth table of related compound statement is as follows:

2) Construct the truth tables for any ¬( p ∨ q) and ¬p ∧¬q and compare them.

Solution:

First, we make a table that displays all the possible combinations of truth values for p and q , 

then we add two columns and put p ∨ q and ¬( p ∨ q) into its top cell and then start calculations.

Note: Ambiguity of “or” in English language

In natural language “or” has two meanings:

Inclusive or: where p ∨ q is true if either p or both are true.

Example: Numbers or measurements may be taken as prerequisites for geometry.

Meaning: take either one but you may also take both.

Exclusive or: denoted as ⊕: where p⊕q is true if either p or q but not both are true.

Example: “You will be paid money or a computer”.

Meaning: do not expect to get both. Bellow is the table for p⊕q

APPLICATION ACTIVITY 2.3.3

1.Translate each of the following compound statements into symbolic form

i) Bwenge reads News Paper or Mathematics book.

ii) Rwema is a student-teacher or not a book seller.

2. Suppose that p is a false statement, and q is a true statement.

a) What is the truth-value of the compound statement¬p ∨ q ?

b) What is the truth-value of the compound statement p ∨¬q ?

3. Let p and q be two propositions. Construct the truth table of

p ∨ q ; p ∨¬q ; ¬( p ∨ q) ∧(¬p ∨¬q) and (¬q ∨ p) ∧(¬p ∨ q)

2.3.4 Conditional statement

ACTIVITY 2.3.4

1) Given the following conditional statement: “If Samantha’s health is good, then she will go to the party”. 

Discuss and identify the hypothesis (premise) and conclusion (consequent) of the conditional statement.

2) Complete these sentences referring to the conditional statement given in question one.

If Samantha’s health is not good, ...

Samantha will go to the party, ...

Samantha will not go to the party, ...

Content summary

A conditional statement is a logic statement used when the statement is in the

form “if ...then ....” It can be written as "If p then q" or p→q ,(or p⇒q ) read as pimplies q .

In this case the proposition p is called antecedent, hypothesis or premise while

the proposition q is the conclusion or the consequent. In language, p can be

called the first clause and q is the second clause.

The conditional p⇒q can also be read:

•  If p, then q.
•  q follows from p.
• q if p.
• p only if q.
•  p is sufficient for q.
•  q is necessary for p.

Let us consider the logical conditional as an obligation or contract, for example:

“If you get 100% on the final, then you will earn an A”

p: If you get 100% on the final, q: you will earn an A.

p⇒q

This shows that F ⇒T does not violate the obligation, the only time the obligation is broken is when T ⇒ F .

 Therefore, the statement p⇒q has the truth value True in all cases except when p is true while q is false.

The related truth table is as follows:

Examples

1.Rephrase the sentence “If it is Sunday, you go to church”, then construct the related truth values.

Solution

Here are some various ways of rephrasing the sentence:

•  “You go to church if it is Sunday.”
•  “It is Sunday only if you go to church.”
•  “It can’t be Sunday unless you go to church.”

Let denote the given statements as: p: It is Sunday, q: you go to church.

The statement If it is Sunday, you go to church is symbolized as follow: p⇒q .

1. Construct the truth value of the following statement:

“If either John takes Calculus or Betty takes Sociology then Peter will take English.”

2. Let denote the statements as:

p: John takes Calculus, q: Betty takes Sociology and r: Peter takes English.

The given statement can be symbolized as follow: ( p ∨ q)⇒r

Use different related sentences and construct the truth table for ( p ∨ q)⇒r .

Solution

Converse, Contrapositive and Inverse of a conditional propositional

a) Converse

The implication obtained by interchanging the antecedent and the consequent

of an implication is called the converse.

Thus, the converse of P ⇒ Q is Q ⇒ P.

Example: If you are a student, then you should study. The converse is “if you study, then you are a student” this is not true.

b) Contra-positive

Given the conditional p⇒q , the implication ¬q⇒¬p is called the contrapositive of p⇒q .

Example:

If it is a PC, then it is a computer, the contrapositive is “if it is not a computer, then it is not a PC”. This is true.

c) Inverse

Given the conditional p⇒q , the implication ¬p⇒¬q is called its inverse. It is

obtained by negating the antecedent and negating the consequence.

Example:

If it is a PC, then it is a computer; the inverse is “If it is not a PC, then it is not a computer”. This is false.

Note:

i. A conditional statement is logically equivalent to its contrapositive. A

conditional statement can be replaced with its contrapositive and keeps its truth value.

ii. Converse: p⇒q does not necessary imply q⇒ p .

iii. Inverse: p⇒q does not imply 

APPLICATION ACTIVITY 2.3.4

1) Using the statements p :Mico is fat and p :Mico is happy

Assuming that “not fat” is thin, write the following statements in symbolic form

a) If Mico is fat then she is happy.

b) Mico is unhappy implies that Mico is thin

2) Write the following statements in symbolic form and their truth values

a) If n is prime, then n is odd or n is 2.

b) If x is nonnegative, then x is positive or x is 0.

c) If Tom is Ann’s father, then Jim is her uncle and Sue is her aunt.

3) Consider the conditional proposition: “If a number is a multiple of

10, then the number is a multiple of 5”.

(a) What is the truth value of that proposition?

(b) Deduce the converse of the proposition and the related truth value.

(d) Deduce the contrapositive of the proposition and the related truth value.

(f) Write its inverse and the related truth value.

4) Write the converse, inverse and contrapositive of the false conditional statement below and determine whether each of the 

    statements found is true or false.

“If x is an even number, then the last digit of x is 2.

2.3.5 Bi-conditional statements

ACTIVITY 2.3.5

1. Let p be the statement: “Anne Maria is intelligent” and q be the statement: “Anne Maria is hard working”

a. Express p⇒q and its truth value.

b. Express q⇒ p and its truth value

c. Give the truth value of ( p⇒q) ∧(q⇒ p) .

2. Let r be the statement: “ 7 = 7 ” and s be the statement: “5 = 3”

a. Give the truth value of r ⇒ s

b. Give the truth value of s⇒r

c. Give the truth value of (r ⇒ s) ∧(s⇒r ).

Content summary

Let us consider the proposition p: Two lines are perpendicular, q: Two lines form a right angle.

We have: p⇒q : If two lines are perpendicular, then the two lines form a right angle.(True)

The converse is q⇒ p means: if two lines form a right angle, then the two lines are perpendicular. (True).

We see that a statement and its converse are both true statements; that is p⇒q and q⇒ p are both true.

Generally p⇒q is not the same as q⇒ p . It may happen, however, that both

p⇒q and q⇒ p are true. The statement p⇔q is defined to be the statement ( p⇒q) ∧(q⇒ p) .

For this reason, the double headed arrow p⇔q is called the bi-conditional.

The bi-conditional p⇔q , which we read “p if and only if q” or “p is equivalent

to q” is true if both p and q have the same truth values and false if p and q have opposite truth values.

Examples

Let denote the statements as:

p : The number is divisible by 3

q : The sum of the digits forming the number is divisible by 3.

The compound statement “The number is divisible by 3 if and only if the sum of

the digits forming the number is divisible by three”;

This means that If the sum of the digits forming the number is divisible by 3,

then the number is divisible by 3 and if the number is divisible by 3, then the

sum of the digits forming the number is divisible by 3.

Symbolically p⇔qmeans ( p⇒q) ∧(q⇒ p) .

3) Given the statements: p: Two lines are perpendicular and q: Two lines form a right angle. 

Formulate 4 different compound statements and their truth values and deduce the truth table of p⇔q .

Solution

One of these compound statements is: “If two lines are perpendicular, then the two lines form a right angle”. 

Make others and you will see the following table.

Equivalent statements

Let us compare the truth values of ¬( p ∨ q) and ¬p ∧¬q .

a) Considering the truth values of ¬p ∧¬q , we can determine the case in which this statement ¬p ∧¬q is true.

The first two columns of the truth table give all possible combinations of the truth values of p and q , and the second two columns of the truth table are merely negations of the first two columns. The statement ¬p ∧¬q is the conjunction of two statements ¬p and ¬q . Therefore, the only case in which ¬p ∧¬q is true is when both A and b are false.

b) The truth values of ¬( p ∨ q) can be summarized in the truth table:

Like the previous question, the truth table consists of four rows. 

The third column of the truth table gives the truth values for the disjunction p ∨ q . 

The fourth column gives the truth values for the negation of disjunction. Therefore, 

¬( p ∨ q) is true only when both p and q are false.

Comparing the truth values of ¬( p ∨ q) and¬p ∧¬q in the last columns of the

two tables, we find that their truth values are identical.

Whenever two statements have the same truth values, the statements are said to be logically equivalent.

 The symbol of equivalent statements is ≡ . Thus we can write ¬( p ∨ q) ≡ ¬p ∧¬q .

In spoken language, the equivalence of two propositions means they are similar, 

when you express one you have the same idea with he who expressed the other.

Example:

Use a truth table to show that the conditional p⇒q and its contrapositive ¬q⇒¬p are logically equivalent.

Solution

Let us construct the truth table containing both p⇒q and its contrapositive ¬q⇒¬p .

The order of ¬q and ¬p when organizing the columns of the table must be respected.

It is clear that p⇒q and ¬q⇒¬p have the same truth values in the third and the sixth columns.

Note: Two propositions differ if you can find at least one row of the truth table where values differ.

APPLICATION ACTIVITY 2.3.5

1) Suppose that r is a false statement, and s is a true statement.

a) What is the truth-value of the compound statement (¬r )⇔ s ?

b) What is the truth-value of the compound statement r ⇔(¬s) ?

c) What is the truth-value of the compound statement r ⇔ s ?

d) What is the truth-value of the compound statement¬(r ⇔(¬s)) ?

2) Construct the truth table for

a) p↔q and ( p→q) ∧(q→ p) b) p↔q and (¬p ∨ q) ∧(¬q ∨ p)

c) ¬( p↔q) and ( p ∨ q) ∧¬( p ∧ q)

d)¬( p↔q) and ( p ∧¬q)∨ (¬p ∧ q)

3) Show that neither the converse nor the inverse of an implication are equivalent to the implication.

4) Formulate a conditional statement made by two compound statements p and q. 

Use that statement to express different forms of ¬p ∨ q and p⇒q and show that ¬p ∨ q and p⇒q are equivalent.

 Relate to the daily life situation

2.4 Tautology and contradiction

ACTIVITY 2.4

Let p and q be two propositions. Construct the truth table of

1) p ∨¬p 2) p ∧(¬p)

3) ¬p ∧( p ∧ q) 4)¬( p ∧ q)∨ ( p ∨ q)

Do you find a column in which the truth value is always true or always false?

Content summary

A tautology is a compound statement that is always true regardless of the truth values of the individual statements substituted for its statement variables.

Example

• The statement “The main idea behind data compression is to compress data” is a tautology since it repeats the same and it is always true.

• The statement “I will either get paid or not get paid” is a tautology since it is always true

If you are given a statement and want to determine if it is a tautology, then all you need to do is construct a truth table for the statement and look at the truth values in the final column.

If all of the values are T (for true), then the statement is a tautology.

The statement “I will either get paid or not get paid” is a tautology since it is always true. We can use p to represent the statement “I will get paid” and not p (written ¬p) to represent “I will not get paid.”

p : I will get paid

¬p : I will not get paid

So, p ∨ (¬p) : I will either get paid or not get paid

A truth table for the statement would look like:

Looking at the final column in the truth table, one can see that all the truth values are T (for true). 

Whenever all of the truth values in the final column are true, the statement is a tautology. 

So, our statement ‘I will either get paid or not get paid’ is always a true statement, a tautology.

A contradiction is a compound statement that is always false regardless of the

truth values of the individual statements substituted for its statement variables.

Example

1) The statement “I don’t believe in reincarnation, but I did in my past life”, is always false.

2) The statement p ∧(¬p) is always false, because p and ¬p cannot both be true.

APPLICATION ACTIVITY 2.4

1) From the following compound statements, indicate which is tautology, contradiction or neither

a)    p ∧¬( p ∧ q)                       b)¬q ∧(q ∧ r )

c)   ( p ∧ q) ∧¬( p ∨ q)           d) ( p ∨ r )∨¬r

2) In your own words, formulate 2 propositional statements expressing the tautology and 2 propositional statements expressing the contradiction. and show where these tautology or contradiction can be avoid in every day practice of argumentation.

2.5 Quantifiers

2.5.1 Universal quantifier

ACTIVITY 2.5.1

1) Let A = {1,2,3,4,5}. Determine the truth value of each of the following

statements:

a) For all x∈ A, x + 3 < 9

b) For all x∈ A, x + 3 ≤ 7

2) Determine the truth value of the following statement where U = {1,2,3}is the universal set:

There exist x∈U such that x + 3 ≤ 5

3) What is the truth value of p, q if p: “All men eat banana”, q: All men are mortal.

Content summary

Let p(x) be a propositional function defined on a set A. Consider the expression

(∀x∈ A) p(x) or ∀x p(x)which is read by “For every x in A , p(x) is a true statement” or, simply, “For all x , p(x) .”

The symbol universal quantifier “For all” symbolized by ∀ is read by “for all” or “for every” is called the universal quantifier.

The statement (∀x∈ A) p(x) means that the truth set of p(x) is the entire set A .

Example:

All men are mortal. This is true if for every man, dying is applicable.

The expression p(x) by itself is an open sentence or condition and therefore has no truth value.

 However, (∀x∈ p(x) , that is p(x) proceeded by the quantifier ∀ , does have a truth value which follows from the equivalence.

Specifically:

1 Q : if {x / x∈ A, p(x)} = Athen ∀x p(x) is True; otherwise ∀x p(x) is False.

Example

APPLICATION ACTIVITY 2.5.1

1. Given the statement p: “for every real number x , x + y >10."discuss the values of y which make p true.

2. Give the truth value for the following statements:

a) Some rectangles are squares

b) All squares are rectangles

c) Every language student-teacher must take a logic mathematics course.

d)

2.5.2 Existence quantifier

ACTIVITY 2.5.2

1) Let A = {1,2,3,4,5}. Determine the truth value of each of the following

statements:

a)There exist x∈ A, x + 3 =10

b) For some x∈ A, x + 3 < 5

2) Let A = {1, 2,3,...,8,9,10}. Consider each of the following sentences;

determine the set of y for which the following statement is true. Such set is called a truth set.

        (a) (∀x∈ A)(∃y∈ A)(x + y <14)

        (b) (∃y∈ A)(x + y <14) .

Content summary

Let p(x) be a propositional function defined on a set A . Consider the expression: there exist some x∈ A that satisfy p(x) . 

This can be written as (∃x∈ A) p(x) or ∃x, p(x) or “There exists an x in A such that p(x) is a true

statement” or, simply, “For some x , p(x) .” The quantifier “there exist” or “for some” or “for at least one” symbolized by ∃ is an existential quantifier. The statement (∃x) p(x) is read as “there exists an x such that p (x) is true”.

Specifically:

Q : if {x / p(x)} ≠ 0 then ∃x p(x) is True; otherwise ∃x p(x) is False.

Example

c) Some men do not have wives, is true since some men such as Priests remain single.

d) There exist natural number between 3 and 6. This is statement is true because {4,5} exists.

e) Some student teachers teach at university. This is false because the set of such students is empty.

APPLICATION ACTIVITY 2.5.2

1. Translate the following into symbolic form:

a) Somebody cried out for help and called the police.

b) Nobody can ignore her.

2. Consider the predicates: r ( x) : x − 7 = 2 and s ( x) : x > 9 . If the universe of discourse is the real numbers, 

give the truth value of propositions: (∃x) s ( x) ∧¬(∀x)r ( x)

3. Formulate different statements with existential quantifiers and give their truth values.

2.5.3 Negation of quantifiers

ACTIVITY 2.5.3

Negate the following statements

1. All grapefruits have red colour.

2. Some celebrities are beautiful.

3. No one weighs more than five hundred Kilograms.

4. Some people are more than 2m tall.

5. All snakes are poisonous.

6. Some mammals can stay under water for two days without surfacing for air.

7. All birds can fly.

Consider the statement: “All dogs have tails”. Its negation is “Not all dogs have tails” which means that:

 “There is at least one dog that does not have a tail.”

Let us express these ideas symbolically:

Examples

1. In statements that involve the words “all, every, some, none or no”, forming the negation is not as easy. 

The following table shows examples

This shows that:

“Negating a universally quantified statement changes it into an existentially quantified statement and vice-versa with the

   part of the statement after the quantifier becoming negated.

2. Let p (x)mean “country x has a president.” Interpret the following statements.

Establish the equivalence of their negation.

Solution

a) Since p(x)mean “country x has a president, thus ∀x, p ( x) means “all countries have president” 

The negation is ,that is interpreted as “it is not true that all counties have presidents.”

This sentence can be express as “there exist a country without a president” and symbolically it is written as

b) ∃x, p ( x) means “there is a country with a president.”

Negating the given statements, we get

This sentence is equivalent to” all countries have not a president” symbolically

These laws can be summarized as follow:

“Negating a universally quantified formula changes it into an existentially

quantified formula and vice-versa with the part of the formula after the quantifier becoming negated.”

The basic forms for negative statements that involve “all, every, some, none or no” can be summarized as follows:

APPLICATION ACTIVITY 2.5.3

Negate each of the following statements and write the answer in symbolic form:

1. Some students are mathematics majors.

2. Every real number is positive, negative or zero.

3. Every good boy does fine.

4. There is a broken desk in our classroom.

5. Lockers must be turned in by the last day of class.

6. Haste makes waste.

2.6 Applications of logic in real life

ACTIVITY 2.6

It is known that you must have the correct form and true premises to

reason deductively toward a true conclusion.

The following are examples of arguments that need true conclusions.

Try to conclude:

a) If you live in Nyarugenge, then you live in Kigali.

If you live in Kigali, then you live in Rwanda.

Therefore, ...

b) If a triangle is isosceles, then it has two equal sides.

If a triangle has two equal sides, then it has two equal angles.

Therefore, ...

c) If you study the whole student book, then you will pass the exam.

You study the whole student book.

Therefore, ...

Content summary

“A syllogism is form of argumentation in the deductive way that consists of two statements or premises, and a logical conclusion drawn from them. These premises are usually articles of faith, laws, rules, definitions, assumptions, or commonly accepted facts.” In the following paragraphs, we discuss three types of syllogisms: hypothetical syllogism, affirming the antecedent, and denying the consequent.

2.6.1 Hypothetical Syllogism

If A, B, and C represent statements, a hypothetical syllogism is constructed from the statements, the first two lines being the premises and the third being the conclusion. The hypothetical syllogism can be written in three different ways:

∴ Means therefore

In the hypothetical syllogism, the argument is correct even when one or both of the premises are false. 

The truth or falsehood of the premises does not affect the logic of the argument.

 Logic deals with the relationship between premises and conclusion, not the truth of the premises.

To say that a deductive argument is correct means that the premises are related to the conclusion in such a way that, if the premises are true, the conclusion must be true. A conclusion cannot be false if the logical form is correct and the premises are true.

Example

a) If you live in Rwamagana, then you live in Eastern province.

If you live in Eastern province, you live in Rwanda.

Therefore, if you live in Rwamagana, then you live in Rwanda.

Solution: It is the hypothetical syllogism.

b) Is the following argument a hypothetical syllogism? Why or why not?

If you have a party, you should invite your friends.

If you are graduating from college, you should invite your friends.

Therefore, if you are having a party, you are graduating from college.

Solution:

The argument is not a hypothetical syllogism. The premises do not link properly. The conclusion

of the first premise should be the hypothesis of the second premise, and no logical

rearrangement can accomplish the proper linking of the statements.

a) Even though the conclusion of this argument is true, explain why the following argument is a poor one.

If you are over 18 years old, then you can read.

If you can read, you can vote.

Therefore, if you are over 18 years old, then you can vote.

Solution

The argument has the form of hypothetical syllogism, so it is a correct argument.

However, it is a poor argument, since neither of the premises is true, the argument does not actually prove its conclusion.

2.6.2 Affirming the Antecedent

If A and B represent statements, an argument that affirms the antecedent has the following form.

Major premise: A⇒ B

Minor premise: A

Conclusion: ∴B

The major premise is a conditional statement A⇒ B . The minor premise states

that the hypothesis of the major premise is true or has occurred.

This is called affirming the antecedent.

Example

If I study for 6 hours, I will pass the exam.

I studied for 6 hours.

Therefore, I will pass the exam.

Solution:

This classical argument is another example of affirming the antecedent.

All men are mortal.

Makuza is a man.

Therefore, Makuza is mortal.

This can be written so that the correct form is apparent.

If one is a man, then one is mortal.

Makuza is a man.

Therefore, Makuza is mortal.

If an argument has the correct form, it is a logical argument. 

However, if it is to be a convincing argument with a true conclusion, its premises must also be true. 

You can affirm the antecedent to reason deductively if the argument has the correct form and true premises.

Example

Is the following argument a good one? Explain.

If you want to run a marathon, then you should train for the race.

Mukamurenzi wants to run a marathon.

Therefore, Mukamurenzi should train for the race.

Solution:

The argument has the correct form for affirming the antecedent. If we take its first premise as true because of commonly accepted notions about the physical stamina needed to run a marathon, the argument is a good one.

Even though the argument has the correct form of an argument using the technique of affirming the antecedent, the major premise is not true. Thus, it is a correct argument but it does not arrive at a true conclusion. 

You need both the correct form and true premises to ensure true conclusions.

2.6.3 Denying the Consequent

If A and B represent statements, an argument that denies the consequent has

the following form:

Major premise: A⇒ B

Minor premise: ~A

Conclusion: ∴~ A

Examples

1. If Edna is at the school, then she has notebooks.

Edna does not have notebooks.

Therefore, Edna is not at the school.

2. If you pay the bill on time, then you are not charged a penalty

You are charged a penalty.

Therefore, you did not pay the bill on time.

The major premise is a conditional statement. The minor premise is a denial (negation) of the consequent (conclusion) of the conditional statement.

 For this reason, this argument is called denying the consequent.

This form of argument is based on the contra -positive principle in which the statement A⇒ B is logically equivalent to ~ B⇒~ A . We can see that this form of argument is correct by observing that it is really an application of affirming the antecedent.

Major premise: A⇒ B is equivalent to ~ B⇒~ AMinor premise: ~ B is

equivalent to ~ B

Conclusion: ~A is equivalent to ~ A

Example

Is the following argument a good one? Explain.

If a number is not positive, then the number is negative.

Zero is not negative.

Therefore, zero is positive.

Solution:

The argument has the form of an argument using denying the con-sequent, so it is a correct argument. 

However, its first premise is not true, since if a number is not positive, it could be either negative or zero. 

Thus, the argument is faulty.

We can also make correct arguments from premises that do not at first glance 

seem to be one of our standard logical forms, as in the next example.

Example

Construct a logically correct argument from the following premises:

If P, then ~ Q .

If ~ R , then Q.

If R, then ~ S .

Solution:

To have the correct form of the hypothetical syllogism, the conclusion of one statement must be the hypothesis of the next statement. Since we know that if a statement is true, its contrapositive is true, we can use that principle on the second premise, that is, “If ~R , then Q” implies “If ~ Q , then R.”

If P, then ~ Q .

If ~ R , then Q.

If R, then ~ S .

This can be reformulated in the following way:

If P, then ~ Q .

If ~ Q , then R.

If R, then ~ S

∴ if P then ~ S

Thus, the conclusion of this argument is: if P then ~ S

Even though you may use the contrapositive statement in a logical argument, do not use the inverse or converse.

APPLICATION ACTIVITY 2.6

1. What is wrong with the following argument?

All good chess players wear glasses.

Sylvia is a good chess player.

Therefore, Sylvia wears glasses.

2. Determine whether or not the following arguments are correct. For those are not correct,

(a) Explain what is wrong with the argument;

(b) Change the minor premise and make a correct argument.

i. When it is midnight, I am asleep.

I was asleep.

Therefore, it was midnight.

ii. All Rhode Island Red hens lay brown eggs.

My hen, Motopa, is a Rhode Island Red.

Therefore, Motopa lays brown eggs

iii. If ABCD is a square, it has four sides.

If it has four sides, then it is a quadrilateral.

Therefore, if ABCD is a square, it is a quadrilateral.

iv) If a triangle is equilateral, then it has three equal sides.

ABC does not have three equal sides.

Therefore, ABC is not equilateral.

2.7 END UNIT ASSESSMENT

1. Which of the following sentences are propositions?

a) Pretoria is the capital of South Africa

b) Is this concept important?

c) Wow, what a day!

2. Find the negation of the proposition “Today is Monday”

3. Find the conjunction of the propositions p and q, where p is the

proposition “Today is Sunday” and q is the proposition “The moon is made of cheese”.

4. Find the disjunction of the propositions p and q, where p is the proposition “Today is Sunday” and q:

 “The moon is made of cheese”.

5. How many rows, not counting the top one, are needed to construct the truth table for a compound

 statement made from ⁿ statements?

6. Show that  is a tautology.

7. Find the truth value of the bi-conditional “The moon is made of cheese if and only if 1=2”.

8. Construct the truth table for the proposition (¬p→q) ∧ r

9. In the question below are given three statements, followed by conclusions: I, II, III, IV. You have to take the given statements

to be true even if they seem to be at variance from commonly known facts. Read the conclusions and then decide which of the

given conclusions logically follows from the given statements disregarding commonly known facts.

Statements: Some Cats are Rats. All bats are tables. All Rats are Bats.

Conclusion:

I. Some Cats are bats II. All bats are rats

III. All tables are cats IV. All bats are cats

1. Only I & II follow 2. Only II follows

3. Only I & IV follow 4. None of these