Back to: Further Mathematics SS1

**LOGICAL STATEMENTS**

A logical statement is a declaration verbal or written that is either true or false but not both.

A true statement has a truth value T

A false statement has a truth value F

Logical statements are denoted by letters p, q, r ……

Questions, exclamations, commands and expression of feelings are not logical statements.

**Example**: Which of the following are logical statements?

i. Nigeria is an African country (Statement)

ii. Who is he? (Not statement)

iii. If I run I shall not be late (Statement)

iv. Japanese are hardworking people (Statement)

v. What a lovely man! (Not statement)

vi. The earth is conical in shape (Statement)

vii. If I think of my family (Not statement)

viii. Take the pencil away (Not statement)

**Evaluation**

State which of the statements is a logical statement

1. Caesar was great leader

2. Oh Mansa Musa, you are wonderful!

3. Is he a serious teacher at all?

4. If 6 is an odd number, then 3 + 5 = 10

5. Stop talking to the boy

6. The Broking House In Ibadan is a magnificent building

**NEGATION**

Given a statement p, the negation of p written ~p is the statement ‘it is false that p” or “not p”

If P is true (T), ~p is false(F)and if P is false(F) ~p is true(T) .

Example I: Let P be the statement ‘Nigeria is a rich country’ then ~p is the statement ‘It is false that Nigeria is a rich country or ‘Nigeria is not a rich country’

Example II: Let r be the statement 3 + 4 = 8 then ~p is the statement 3 + 4 ¹ 8

Example III: Let q be the statement ‘isosceles triangle are equiangular’ then ~q is the statement ‘it is false that isosceles triangles are equiangular or ‘isosceles triangle are not equiangular’.

**Evaluation**

1. Write the negation each of the following statements.

1. It is very hot in the tropics.

2. He is a handsome man.

3. The football captain scored the first goal.

4. Short cuts are dangerous.

2. Write the negation of each of the following avoiding the word ‘not’ as much as possible.

1. He was present in school yesterday.

2. His friend is younger than my brother.

3. She is the shortest girl in the class.

4. He obtained the least mark in the examination.

**Reading Assignment: Further Maths projects Ex. 9a Q 3 – 7.**

**CONDITIONAL STATEMENTS**

Let q stand for the statement ‘Femi is a brilliant student’ and r stand for the statement ‘Femi passed the test’. One way of combing the two statement is ‘If Femi is a brilliant student then Femi passed the test’ or ‘If q then r’

The student ‘If q then r’ is a combination of two simple statements q and r. It is called a compound statement.

Symbolically, the compound statement can be written as follows: ‘If q then r’ as q Þ r

The statement q Þ r is real as

q implies r or

if q then r or q if r

The symbol Þ is an operation. In the compound statement q Þ r, the statement q is called the

antecedent while the substatement r is called the consequence of q Þ r.

**Example**: If q is the statement ‘I am a male’ and r is the statement ‘The sun will rise’

Consider the statements.

a. If I am a male then the sun will rise

b. If I am a male then the sun will not rise

c. If I am not a male then the sun will rise

d. If I am not a male then the sun will not rise

The statement (a), (c) and (d) are all true but b is not true b and c the antecedent is true and the consequent is false.

**CONVERSE STATEMENT**: The statement q Þ p is called the converse of the statement p Þp. e.g. Let p be the statement ‘a triangle is equiangular’ and q the statement ‘a triangle is equilateral’.

The State p Þp means if a triangle is equiangular then u is equilateral.

The statement q Þ p means if a triangle is equilateral then u is equiangular.

**INVERSE STATEMENT**: This statement ~p Þ~ q is called the inverse of the statement p Þ q.

If p is the statement ‘a triangle is equiangular and q is the statement ‘a triangle is equilateral’

the statement ~p Þ~ q is the statement ‘if a triangle is not equiangular then it is not equilateral’.

**CONTRAPOSITIVE STATEMENTS**: The statement ~q Þ~ p is called the contrapositive statement of p Þ q.

If p is the statement ‘I can swim’ and q is the statement ‘I will win’ then the statement ~q Þ~ p is the statement ‘If I cannot swim then I cannot win’.

**Evaluation**

If p is the statement ‘it rains sufficiently’ and q the statement ‘the harvest will be good’ write the symbol of these statements.

(i) If it rains sufficiently then the harvest will be good.

(ii) If it doesn’t rain sufficiently then it doesn’t

(iii) If the harvest is poor then it doesn’t rain sufficiently.

(iv) It doesn’t rain sufficiently.

(v) If it doesn’t rain sufficiently then the harvest will be good.

**IDENTIFICATION OF ANTICEDENCE AND CONSEQUENCE OF SIMPLE STATEMENTS.**

1. Bi-conditional statements

2. The Chain Rule

1. **BICONDITIONAL STATEMENTS : **If p and q are statements such that p Þ q and q Þ p are valid, then p and q

imply each other or p is equivalent to q and we write p Û q. The statement p Û q is called a biconditional

statement of p and q and the statement p and q are equivalent to each other.

p Û q could be read as

q is equivalent to p or

q if and only if p or

p if and only if q or

if p then q and if q then p

A bi-conditional statement is true when two sub-statements have the same truth value.

e.g. If p is the statement ‘the interior angle of a polygon are equal’ and q is the statement ‘a polygon is regular’.

p Þ q is the statement ‘if the interior angles of a polygon are equal then the polygon is regular’.

q Þ p is the statement ‘if a polygon is regular then the interior angles of the polygon are equal’.

p Þ q and q Þ p

p Û q

p and q are equivalent to each other.

Examples: Let p be the statement ‘Mary is a model’

Let q be the statement ‘Mary is beautiful’

Consider these statements.

a. Mary is a model if and only if she is beautiful.

b. Mary is a model if and only if she is ugly.

c. Mary is not a model if and only if she is beautiful.

d. Mary is not a model if and only if she is ugly.

Statements a and d are true b and c the sub-statements have the same truth value. Statements b and c are false because the sub-statements have different truth values.

**2. THE CHAIN RULE: **If p, q and r, are three statements such that p Þ q and q Þ r.

Example I: Consider the arguments

Premise T_{1}: If a student works very hard, he passes his examination

Premise T_{2}: If a student passes his examination he is awarded a certificate.

Conclusion T_{3}: If a student works very hard, he is awarded a certificate.

**SOLUTION**

Let p be the statement “a student works very hard”

Let q be the statement “a student passes his examination”

Let r be the statement “a student is awarded a certificate”

‘The argument has the following structural form.

p Þ q and q Þ r \ p Þ r

This argument follows the chain rule link hence u is said to be valid.

**Example II**: Consider the arguments

T_{1}: Soldiers are disciplined

T_{2}: Good leaders are disciplined men

T_{3}: Soldiers are good leaders.

**SOLUTION**

** **Let p be the statement ‘X is a seller’

Let q be the statement ‘X is a disciplined man’

Let r be the statement ‘X is a good leader’

The argument has the following structural form.

T_{1} : p Þ q

T_{2} : r Þ q

T_{3} : p Þ r

The argument does not follow the format of the chain rule, hence it is not valid.

**Evaluation I**

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

T_{1} : It is necessary to stay healthy in order to live long.

T_{2} : It is necessary to eat balanced diet in order to stay healthy.

T_{3} : It is necessary to eat balanced diet in order to lives long.

**Evaluation II**

(1) Let P be the statement : “He is funny” and q be the statement : “He is serious”. Write each of the following in simple English (i) p v q (ii) p ˄ q (iii) p˄ ~q (iv) ~pv~q

(2) If p and q represent two statements “he is good in physics” and “he is good in mathematics” respectively. write the following in symbolic form; “he is good in physics if and only if he is good in mathematics”.

**General Evaluation**

(1) Find the truth value of these statements.

a. If 11 > 8 then -1< -8

b. If 3 + 4 ¹ 10 then 2 + 3 ¹ 5

(2) Find the values of x satisfying 2^{3x + 1 } – 3 (2^{2x}) + 2^{x + 1} = 2^{x}

(3) Solve the equation 3^{2x} – 30 (3^{x}) + 81 = 0

(4) Solve the simultaneous equations 2x + y = 3; 4x^{2} – y^{2} + 2x + 3y = 16.

**Reading Assignment: F/Maths Project 1 pages 126 – 130 Exercise 9b Q 2, 3 and 4**

**WEEKEND ASSIGNMENT**

P is the statement ‘Ayo has determination and q is the statement ‘Ayo will succeed’. Use this information to answer these questions. Which of these symbols represent these statements?

1. Ayo has no determination.

A. P Þ q B. ~ p Þ q C. ~ p

2. If Ayo has no determination then he won’t succeed.

A. ~p Þ~ q B. p Þ~ q C. p Þ q D. p Þ~ q

3. If Ayo won’t succeed then he has no determination.

A. ~q Þ p B. ~q Þ~q C. ~q Þ p D. q Þ p

4. If Ayo has determination then he will succeed.

A. ~p Þ q B. ~p Þ~ q C. ~q Þ~ p D. p Þ q

5. If Ayo has no determination then he will succeed.

A. ~p Þ q B. ~q Þ~ p C. ~p D. ~p Þ~ q

**THEORY**

1. Write down the inverse, converse and contrapositive of each of these statements.

(i) If the bank workers work hard they will be adequately compensated.

(ii) If he is humble and prayerful, he will meet with God’s favour.

(iii) If he sets a good example, he will get a good followership.

2. Consider the following statements P: Some dogs are tame Q: All tame animals are small.

Which of the following is a valid conclusion from the above statements?

(i) All dogs are tame. (ii) No dog is small. (iii) All small animals are tame. (iv) Some dogs are small.

(v) All tame animals are dogs.

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