LOGICAL REASONING

WEEK 1

SUBJECT: FURTHER MATHEMATICS

CLASS: SS 1

TERM: 3RD

TOPIC: LOGICAL REASONING

CONTENT:

• Statements
• Negation
• Conditional Statement
• Bi-conditional Statement
• The Chain Rule
• Converse, inverse and contra positive of statement.
• Antecedent and consequent
• Connectives for statements including symbols.

Logical Reasoning

Logic is the science of reasoning, and thinking. It is also means ability to draw suitable inferences and conclusions after a persuasive argument. To many, logic may mean the power of expressive language, but a Mathematician breaks down the expressions into propositions from where their ‘truth values’ can be determined.

STATEMENTS

In logic, statement can be define as declaration, verbal or written which is either true or false. Statement can also be referred to as ‘proposition’. However, no statement can satisfy the two conditional (that is true and false at the same time). A statement that is true will have the truth value denoted as. While a statement that is false will have the truth value denoted as

Example 1: The following statements are logical

1. Ghana is a West African Country
2. The earth is spherical in shape.
3. Nigerians are patient people.
4. If I don’t run I shall be late.

Example 2: The following statements are not logical.

1. When I think of my past.
2. Take away the book.
3. Who do you think you are?
4. What a good meal!

Note that, questions, exclamations, commands and expressions of emotions that cannot be assigned truth value of either In logic letters like are used to represents statements.

CLASS ACTIVITY

1. Define logic and state its importance to mathematician.
2. State which of the following are statements in the logical context:
3. Mohammed  Buhari is a great leader.
4. Stop disturbing the girls.
5. Decide whether you are going to the youth meeting.
6. Oh James , Dele, you are superb!
7. If 8 is an odd number, then

NEGATION

If is a statement, the negation of written or is defined as , or “It is false that holds”. We can show the relationship between in a tabular form called truth value table.

Example1: Write the negation of the following statements.

1. The switch is open.
2. The door is locked.

Solution:

1. The switch is open. The switch is not open.
2. The door is locked. The door is not locked or the door is open.

Example2: Give the negation of the following statements.

a. The plane is flying fast.

b. It is a hot day.

Solution:

1. . The plane is not flying fast or The plane is flying slow.
2. . or It is a cold day.

Note:

The solution can be presented in a tabular form also.

CLASS ACTIVITY

Write out the negation of the following propositions avoid the use of the word NOT as much as possible.

1. She was marked absent yesterday.
2. The earth is a global village.
3. His friend is older than my sister.
4. 3 > 5
5. 5 + 2 =7
6. Two sides of a triangle are greater than the third side.

CONDITIONAL STATEMENT

A conditional statement, symbolized by p, q, is an if-then statement in which p is a hypothesis and q is a conclusion. The logical connector in a conditional statement is denoted by the symbol. The conditional is defined to be true unless a true hypothesis leads to a false conclusion.

Using symbols we can write conditional statement (simple or compound). If then means this reads thus:

OROROROR

OR OR.

Example 1

Given two different statements thus:

Abuja is a city in Nigeria

Abuja is a city in West Africa.

We can combine these two propositions to form a compound statement using a conditional word .

If Abuja is a city in Nigeria then Abuja is a city in West Africa. Meaning “if then which forms a “compound proposition”.

Example 2

Determine the truth value of each of the following:

1. If 2 x 3 = 6, then 2 + 3 = 5
2. If 2 x 3 = 6, then 2 + 3 = 7
3. If 2 x 3 = 8, then 2+ 3 = 5
4. If 2 x 3 = 8, then 2 + 3 = 7

Solution

If we assign p to the antecedents and q to consequents in each of the following we find that all the above are true except (ii). Thus (ii) is false while others are true. Even though (iv) has two false statements, the composite p q is true.

CLASS ACTIVITY

Form compound statements using ‘if … then ’

1. P: y = 2

Q:

1. P: A student reads Mathematics

Q: the student reads science

1. P: Damilola is a youth corper

Q: she has a degree

1. Identify the antecedent and consequent in the statement below;

If mathematics teachers work very hard then they will be compensated.

BI-CONDITIONAL STATEMENT (EQUIVALENCE).

Let us consider these statements: the Educators in High School are computer literate AND High School is e-learning.

if the Educators in High School are computer literate then High School is e-learning.

q if High School is e-learning then the Educators in High School are computer literate.[mediator_tech]

Since, and q then which confirm the two statements to be bi-conditional.

Bi-conditional statements are also called equivalent statements. Other words used to read includes:

Truth value table for

Example 1: Find the truth value of the following:

1. If then
2. If then

Solution:

1. is and is ; hence If then is False

is and is ; hence if then is True

Copy and complete the table

P	Q	Q	P ⇒Q	P ˅Q	(P⇒Q)↔(P˅Q)
T	T		T
T	F
F	T				F
F	F			T

THE CHAIN RULE

Given three propositions such that implies and implies r, then implies r. This is called the chain rule. Symbolically, and q then. The statement is called premises while is called conclusion of the argument.

Example 1:In the following argument, determine whether or not the conclusion necessarily follows from the given premises.

1^st premise: All intelligent students do their work satisfactorily.

2^nd premise: All SS 1 students are intelligent.

Conclusion: Therefore, all SS 1 students do their work satisfactorily.

Solution

If p: Students who are intelligent

q: Students who do their work satisfactorily

r: Students who are in SS 1

then 1^st premise

and q2^nd premise

If and q

then (chain rule)

The conclusion follows from the premises.

Example 2: Give an outline of the structural form of the following argument and decide on whether or not the argument is valid.

X: A lazy student begs for marks

Y: A student who begs for marks is a dullard.

Z: Therefore, a lazy student is a dullard.

Solution

If p: A is a lazy Student

q: A is a student who begs for marks

r: A is a dullard

then 1^st premise

and q2^nd premise

If and q

then (chain rule)

The conclusion follows from the premises.

CLASS ACTIVITY

1. Give an outline of the structural form of the following argument and state whether or not it is valid.

any driver who drinks alcohol excessively drives carelessly

any driver who drives carelessly is certain to have an accident.

any driver who drinks alcohol excessively is certain to have an accident.

1. Determine the validity of each of the following arguments.
2. X is a square X is a rectangle

X is a square X is a rhombus.

Therefore, X is a rectangle X is a rhombus.

1. X is a whole number X is an integer

X is an integer X is a rational number.

Therefore, X is a whole number X is a rational number.

CONVERSE STATEMENT:

This is a statement that confirm logical statement to be true through reverse statement.

Example1: Consider this statements; : all dogs are collies” andall collies are dogs. The statement implies and implies i.e. and q. Hence q is the converse statement of

Example 2:Consider this statements; :’If people smoke cigarettes then they will get lung cancer’ and‘People who get lung cancer smoke cigarettes’. The statement implies and implies i.e. and q. Hence q is the converse statement of

INVERSE STATEMENT:

The inverse of conditional statement ‘if then ’ is the conditional statement “ if not then not ” . i.e. the inverse of pq is.

Example 1:Consider this statements;

q :’If people smoke cigarettes then they will get lung cancer’. Hence the inverse statement of i.e. If people do not smoke cigarettes then they will not get lung cancer.

Example 2: Find the inverse of the following statements:

1. If a girl falls then she will break her back.
2. If a student is diligent then he will pass her exams

Solution

1. If a girl does not fall then she will not break her back.
2. If a student is not diligent then he will not pass her exams

CLASS ACTIVITY

1. Write down the inverse of each of the following statements
2. If Mary is a model then she is beautiful
3. If Ibadan is the largest city in the west Africa then it is the largest city in Nigeria
4. If the army misbehaves again he will be demoted
5. Write down the converse of each of the following
6. If he sets a good, he will get a good fellowship
7. If it rains sufficiently then the harvest will be good
8. If the triangles are congruent then the ratios of their corresponding lengths are equal.

CONTRAPOSITIVE STATEMENT:

In logic, contraposition is an inference that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped: the contrapositive of is thus.

This is a situation where two statement that agreed at converse point to give a value, jointly disagree with the statement or negate the statement.

Example 1: all dogs are collies

all collies are dogs.

BUT all dogs are not collies AND all collies are not dogs.

.

Example 2: If all A is B then by contraposition ‘all not B is not A’ or ‘no B is no A’.The table below illustrates this better.

			Converse	Inverse	Contrapositive

 

From this table we can easily infer that if it not true that. Put differently,

Example 3: The contrapositive of “If it is raining then the grass is wet” is “If the grass is not wet then it is not raining.”

Note: As in the examples, the contrapositive of any true proposition is also true.

CLASS ACTIVITY

Objective Test:

1. If , then
2. B. C. D.
3. Which of the following logical statement is /are correct? I. II. III.

A. I and II only B. II only C. I, II and III D. III only

Essay Test:

1. Write down the contrapositive of each of the following:
2. If the politician is popular among the rural folk, he will be elected.
3. If he is kind and merciful, he will receive God’s mercy.
4. If Bala has determination, he will succeed.
5. In order to be a good leader one must be a good follower.
6. Show that the conditional statement and contrapositive statement are true.

Antecedent and Consequent of a Proposition

In logic, an antecedent is that part of a conditional proposition on which the other depends. For example, consider the following

In the use of the implicational signs or symbols statements are defined and differentiated. For instance: The compound statement the statement is called the antecedent while the sub statement is called the consequent of

Consider this statement: If Maryland is in Egypt then it a commercial city. Maryland in Egypt is the antecedent statement while it is a commercial city is the consequent.

Example 1:

The student can solve the problem only if he goes through the worked examples thoroughly.

Antecedent: The student can solve the problem

Consequent: He goes through the worked examples thoroughly

Example 2:

If Dayo is humble and prayerful then he will meet with God’s favour.

Antecedent: Dayo is humble and prayerful

Consequent: He will meet with God’s favour

CLASS ACTIVITY

Identify the antecedent and the consequent parts of this statement: “ If success elude you then you did not work-hard.

Identify the antecedent and the consequent in these implicative statements

1. If I travel then you must teach my lesson
2. If you person well in your examinations then you will go on holidays
3. If London is in Britain then 12 is an even number
4. If the bus come late then I will take a motorcycle
5. If a & b are integers then ab is an integer

Connectives for statements including symbols.

In logic, a logical connective (also called a logical operator) is a symbol or word used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, such that the sense of the compound sentence produced depends only on the original sentences.

The word ‘not’ and the four connectives ‘and’, ‘or’, ‘if … then’, ‘if and only if’ are called logical operators. They are also referred to as logical constants. The symbols adopted for the logical operators are given below.

Logic Operators Symbols Name

‘not’ negation

‘and’ conjunction

‘or’ ˅ disjunction

‘if … then’ conditional

‘if and only if’ ↔ bi-conditional

When the symbols above are applied to propositions p and q, we obtain the representations in the table below:

Logic operation Representation

‘not p’ p or

‘P and q’ p˄q

‘p or q’ p˅q

‘if p then q’ pq

‘p if and only if q’ p↔q

Given two proposition: p: It is raining and q: I am indoors.

Example 1:

Example 2:

Example 3:

Example 4:

Conjunction (or ˄) of logical reasoning: Any two simple statements p,q can be combined by the word ‘and’ to form a compound (or composite) statement ‘p and q’ called the conjunction of p,q denoted symbolically as p˄q.

Example 1. Let p be “The weather is cold” and q be “it is raining”, then the conjunction of p,q written as p˄q is the statement “the weather is cold and it is raining”.

Example 2: The symbol ‘˄’ can be used to define the intersection of two sets A and B as follows;

The truth table for p˄q is given below;

P	Q	p˄q
T	T	T
T	F	F
F	T	F
F	F	F

 

CLASS ACTIVITY

Form compound statements using ‘and’, and express the following compound statements in symbol form.

1. P: It is cold.

Q: It is wet.

1. P:

Q: = -3

Disjunction (or ˅) of logical statements. Any compound statement formed by using the word ‘or’ to combine simple statements is called a disjunction. The symbol ‘˅’ stands for ‘or’.

Examples 1Let p ‘ be “Bola studied Mathematics”, andq’ be “Ngozi studied French”. Then the disjunction of p, q (p˅q) is the statement “Bola studied Mathematics or Ngozi studied French”.

Examples 2

P: The solution of

Q: The solution of -3.

P˅Q: The solution of

The truth table for p˅q is illustrated below

P	Q	p˅q
T	T	T
T	F	T
F	T	T
F	F	F

Express the compound statements in symbolic form.

1. P: is a rational number, q: is an even number.
2. P: the trade union is stubborn, q: the workers strike will soon be ended.

CLASS ACTIVITY

If p, q, r, stand for

P: Birds fly

q: The sky is blue

r: The grass is green

1. Write the sentence that has the same meaning as:
2. c. p ˄ q
3. V q d. (p ˄ ) r
4. Using the proposition p, q and r of the above problem, write the symbolic representation of:
5. The sky is blue and the grass is green.
6. Birds fly or the sky is blue.
7. If the grass is green and the sky is not blue then the birds do not fly.

PRACTICE QUESTIONS

Determine the truth value of the compound statements:

1. (P ˄ Q)
2. p →
3. p ( p →)
4. (p V
5. (p ˄ ) r
6. Show thatp Vis equivalent to p → q

ASSIGNMENT

1. Find the truth value of the following:
2. If 11 > 8 then -1 < -8
3. If 3 + 4 ≠ 10 then 2 + 3 = 5.
4. Give an outline of the structural form of the following argument and state whether or not it is valid:

T₁: It is necessary to stay healthy in order to live long.

T₂: It is necessary to eat balanced diet in order to stay healthy

T₃: It is necessary to eat balanced diet in order to live long.

1. Write down the converse of each of the following statements:
2. If he sets a good example, he will get a good followership.
3. If the triangles are congruent, then the ratios of their corresponding lengths are equal.
4. If a line is perpendicular to a plane, then it is normal to the plane.
5. Write down the contrapositive of each of the following:
6. If the politician is popular among the rural folk, he will be elected.
7. In order to be a good leader one must be a good follower.
8. Given that p and q are statements such that:

p: He is healthy

q: He is neat.

Which of the following represents “he is unhealthy only if he is dirty”?

A. p →

B. →

C. →p

D. p →

1. If P and Q are two logical statements, copy and complete the following truth table

P	Q	P ˅ Q	(P˅ Q)	(P˅ Q) ˄ P	(P˅ Q) ⇒ P

1. Use a truth table to show;

KEY WORDS

Statement, Proposition, negation, connectives/operators, conjunction, disjunction, conditional, bi-conditional, converse, inverse, antecedent, consequent, contrapositive, chain rule