Lesson Notes By Weeks and Term - Senior Secondary School 2

SUBJECT: MATHEMATICS

CLASS:  SS 2

DATE:

TERM: 2nd TERM

REFERENCE BOOKS

• New General Mathematics SSS2 by M.F. Macraeetal.
• Essential Mathematics SSS2 by A.J.S. Oluwasanmi. 

 
WEEK SEVEN

TOPIC: LOGIC

CONTENT

-Meaning of Simple and Compound Statements.

-  Logical Operations and the Truth Tables.

-Conditional Statements and Indirect Proofs.

SIMPLE AND COMPOUND PROPOSITIONS 

A preposition is a statement or a sentence that is either true or false but not both. We shall use upper case letters of English alphabets such as A, B, C, D, P, Q, R, S, …, to stand for simple statements or prepositions. A simple statement or proposition is a statement containing no connectives. In other words a proposition is considered simple if it cannot be broken up into sub-propositions. On the other hand, a compound proposition is made up of two or more propositions joined by the connectives. These connectives are and, or, if …then, if and only if. They are also called logic operators. The table below shows the logic operators and their symbols.

Figure 1

1. The statement ~P is known as the negation of P. thus ~P means not P or ‘it is false that P…’ or ‘it is not true that P…’ 
2. If P and Q are two statements (or propositions), then: 

1. The statement P ⋀ Q is called the conjunction of P and Q. thus, P ⋀ Q means P and Q. 
2. The statement P ⋁ Q is called the disjunction of P and Q. thus, P ⋁ Q means either P or Q or both P and Q. notice that the inclusive or is used. 

1. The statement P Q is called the conditional of P and Q. a conditional is also known as implication P Q means if P then Q or P implies Q. 
2. The statement P Q is called the biconditional of P and Q, where the symbol means if and only if (or iff for short). Thus P Q means P Q and Q P.

The Truth Tables 

The truth or falsity of a proposition is its truth value, ie. A proposition that is true has a truth value T and a proposition that is false has a truth value F. the truth tables for the logical operators are given below. 

Figure 2

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

Recall that other symbols used instead of ~ are P’ or P or ~P.

Figure 3    figure 4   

P⋀Q is true when both             P⋁Q is false when both P and Q are false.

P and Q are true 

    Figure 6

Figure 5

P Q is false when P is             P Q is true when both P and Q are 

true and Q is false             either both true or both false. 

Example 1

Translate the following into symbols and then determine which statements are true or false. 

1. -5 < 8 and 2 < - 50 
2. 4 right angles = 360o or opposite angles of any quadrilateral and supplementary. 
3. If a person is 20 years old, then the person is a teenager. 
4. 2x – 5 = 9 if and only if x = 7. 

Solution 

1. Let P = (-5 < 8); Q = (2 <  -50)

P - -5 < 8 is true (T)

Q = 2 < -50 is false (F)

∴symbolic for: P  ⋀ Q is false 

(see 2nd row of fig 3)

1. Let P = (4 right angles = 360o)

Q (opposite angles of any quadrilateral are supplementary).

P is true (T) and Q is false (F)

∴ P⋁Q is true (see 2nd row of fig 4) 

1. Let P = a person is 20 years old. 

Q = a person is a teenager. 

P is T and Q is F

∴ PQ is false (see 2nd row of fig 5) 

1. 2x – 5 = 9 if and only if x = 7

Let P = (2x – 5 = 9) and Q = (x = 7)

When x = 7, 2x – 5 = 2 X 7 – 5 

            = 14 – 5 = 9 (T)

Both P and W have the same T values. 

∴ P⟺Q is true (see 1st row of fig 6) 

Converse, Inverse and Contrapositive of Conditional Statement 

Converse statement

The converse of the conditional statement ‘if P then Q’ is the conditional statement ‘if Q then P’, i.e. the converse of P Q is Q P. 

Inverse statement 

The inverse of the conditional statement ‘if P then Q’ is the conditional statement ‘if not P then not Q’.

i.e. the inverse of P Q is P ⇒∼Q.

Contrapositive statement 

The converse of the conditional statement ‘if P then Q’ is the conditional statement ‘if not Q then not P’.

i.e. the contrapositive of P Q is P ⇒∼P.

Example 

Give the (a) converse (b) inverse 

(c) contrapositive of the following:

(i) If 9 < 19, then 8 < 5 + 6.

(ii) if two triangles are equiangular, then their corresponding sides are proportional. 

Solution 

1. (i) if 8 < 5 + 6 9 < 19. 

(ii) If two triangles have their corresponding sides proportional, then they are equiangular. 

1. (i) if 9 ≮19 8 ≮ 5 + 6.

(ii) If two triangles are not equiangular, then their corresponding sides are not proportional 

1. (i) if ≮ 8 + 6 9 ≮ 19

(ii) if two triangles do not have their corresponding sides proportional, then they are not equiangular 

LOGICAL OPERATIONS AND TRUTH TABLES

Example 

Construct the truth tables for the following: 

(P ⋁∼Q) (P ⋀∼Q)

Solution 

Method 1 

Explanation 

Since there are two variables, P and Q, we will have 4 rows. 

1. P column: Enter two T’s, then two F’s 
2. Q column: Enter one T, then one F.
3. P column: Enter the negation of P.
4. Q column: Enter the negation of Q. 
5. Fill in the truth values of ∼P ⋁ ∼Q.

∼P ⋁ ∼Qis false when both  P and ∼Q are false according to the table for V.

1. Fill in the truth values of P ⋀∼Q is true when both P and Q are true.
2. Fill in the truth value of 

(P V∼Q) (P VQ)

(P V∼Q) (P VQ) is false when (P V∼Q) is true and (P V∼Q) is false. 

Method 2 

      (1)      (2)       (3)        (4)        (5)

Explanation 

Enter P and Q columns as usual. 

• fill in the truth values of ∼P.

• Fill in the truth values of Q.

• Fill in the truth values of (P ⋀Q)

• Fill in the truth values of (P ⋀Q).

• Now consider the implication () as a whole. 

Note:     The columns of the truth table are completed in the indicated order

Tautology and Contradiction 

When a compound proposition is always true for every combination of values of its constituent statements, it is called a tautology. On the other hand, when the proposition is always false it is called a contradiction.

Example 

Construct the truth tables to show that: 

(a) P ⇔∼ (P) is a tautology 

(b) (P ⋀Q) [(P) V (Q)] is a contradiction. 

Solution 

The truth table of P ~(~P) is always T, so it is a tautology

        (1)    (5)      (2)    (4)       (3)

Column (5) shows that the truth table of (P ⋀ Q) [(~P) V (~Q)] is always F, so it is a contradiction. 

Example 

Find the truth values of the following when the variables P, Q and R are all true. (a) ~P ⋀~Q     (b) ~(P ⋀~Q) V~R

Solution 

1. ~P ⋀~Q 

Substituting the truth values directly into the statement ~P ⋀~Q, we have ~T ⋀~T. 

But ~T is the same as F. 

∴    ~T ⋀~T gives     F ⋀ F 

Simplify the disjunction:    F 

∴    The compound statement ~P ⋀~Q is false. 

1. ~(P ⋀~Q) V~R 

Substituting the truth values: ~(T ⋀~T) V~T 

Within brackets, negate:     ~(T ⋀ F) V~T 

Simplify brackets:     ~ F V~T

                T V F 

Simplify disjunction: T 

∴    ~(P ⋀~Q) V~R is true. 

Example 

Determine the validity of the argument below with premises X1 and X2 and conclusion S. 

X1 = All doctors are intelligent 

X2: Some Nigerians are doctors 

S: Some Nigerians are intelligent 

In the Venn diagram 

E= {all people}

Let I = {intelligent people}

N = {Nigerians}

D = {doctors} 

        I              N   

          D               

                    E

The structure of the argument is shown in figure above. The shaded region represents N ⋂ I, those Nigerians who are intelligent. The conclusion that some Nigerians are intelligent therefore follows from the premises, and the argument is valid. 

Example 

In the following argument, find whether or not the conclusion necessarily follows from the premise. Draw an appropriate Venn diagram and support your answer with a reason. 

London is in Nigeria 

Nigeria is in Africa. 

Therefore London is in Africa 

The figurebelowshows the data in a Venn diagram. 

                Nigeria 

                    Africa 

                London 

From the figure above, the conclusion follows from the premises, L N and N A. the argument is therefore valid. 

Notice, however, that the conclusion in untrue because the first premise ‘London is in Nigeria’ is untrue. Therefore, we may have an argument that is valid but in which the conclusion is untrue. 

THE CHAIN RULE 

The chain rule states that if X, Y and Z are statements such that X Y and Y Z, then X Z. a chain of statements can have as many ‘links’ as necessary. Example 5 is an example of the chain rule. 

When using chain rule. It is essential that the implication arrows point in the same direction. It is not of much value, for example, to have something like X Q R because no useful deductions can be made from it. 

Example 

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

All drivers are careful. (1st premise)

Careful people are patient (2nd premise)

Therefore all drivers are patient (conclusion)

If    D: people who are drivers 

    C: people who are careful 

    P: people who are patient  

Then    D C        (1st premise)

And    C P        (2nd premise)

If    D C    and    C P

Then     D P        (chain rule)

The conclusion follows from the premises. 

Example 

Determine the validity of each of the proposed conclusions if the premises of an argument are 

X: Teachers are contented people. 

Y: Every doctor is rich 

Z: No one who is contented is also rich. 

Proposed conclusions 

S1: No teacher is rich 

S2: Doctors are contented people 

S3: No one can be both a teacher and a doctor.

Let     C = {contented people}

T = {teachers}

D = {doctors}

R = {rich people}

The figure below is a Venn diagram for the premises.  

              T            D

              C                    R   

From the figure, the following conclusions can be deduced. 

1. S1 is true, i.e. no teacher is rich. (T ⋂ R = ∅)
2. S2 is false, i.e. doctors are contented people is false. (D ⋂ C = ∅)

iii. S3 is true, i.e. no one can be a teacher and a doctor. (T ⋂ D = ∅)

CONDITIONAL STATEMENTS AND INDIRECT PROOFS.

Another method we can use to determine the validity of arguments especially the more complex ones is to construct the truth tables as will be seen in the following examples. 

Example 1 

Write the argument below symbolically and determine whether the argument is valid. 

1st premise:    if tortoises eat well, then they live long 

2nd premise:    Tortoises eat well. 

Conclusion:    Tortoises live long. 

Solution 

To determine the truth value, the steps are: 

1. Write the arguments in symbolic forms. 

Let P = ‘tortoises eat well’

    Q = ‘they live long’. 

1st premise becomes P Q. 

2nd premise is P and the conclusion is Q. 

∴the argument is written as follows: 

P Q     (if P happens, then Q will happen)

PQ        (P happens)

        (Q happens) 

1. From the conjunction of the two premises. (P Q) ⋀P
2. Let the conjunction in (2) implies the conclusion Q. i.e. [(P Q) ⋀ P] Q

Since the compound statement 

[(P Q) ⋀ P] Q is always a tautology, (i.e. has a truth value T), the argument is valid. This type of argument is called direct reasoning or modus ponems 

Example 2

Determine whether the following argument is valid. 

If you study this book, then you will pass WAEC. 

If you pass WAEC, then you will go to university

Therefore, if you study this book, then you will to go university. 

Solution 

1. Let P: you study this book

Q: you will pass WAEC. 

R: you will go to university.

If you study this book, then you will pass WAEC becomes P Q. 

If you pass WAEC, then you will go to university becomes Q R.

Therefore, if you study this book, then you will go to university becomes P R. 

The above may be written as follows: 

1st premise:    P Q 

2nd premise:     Q R 

Conclusion:    P R 

1. From the conjunction of the premises as (P Q) ⋀ (Q R)

1. Let the conjunction implies the conclusion implies the conclusion. i.e.

[(P Q) ⋀ (Q R)] (P R)

P      Q    R    [(P Q) ⋀ (Q R)] (P R)

T    T    T    T    TTTT

T    T    F          T      F       F        T         F

T        F    T          F    F       T        TT

T        F    F    F    F       T        T        F

F        T    T    T    TTTT

F        T    F          T      F       F        T        T

F    F    T    T    TTTT

F    F    F          T    TTTT

                  (1)      (3)     (2)      (5)      (4)

Column (5) shows that the compound statement [(P Q) ⋀ (Q R)] (P R) is always tautology. Therefore, the argument is valid. This type of argument is called transitive reasoning or chain rule or the rule of syllogism. 

Note: there are other forms of valid arguments which you can investigate on your own. 

EVALUATION 

1. Choose a letter to represent each simple proportion and then write the following in symbols. 

1. David is a lazy student and he refuses to do his home work. 
2. If a number is divisible by 2, then it is an even number.
3. If the soup does not contain adequate ingredients, then the soyp will not taste nice.  

1. Determine the truth values of the following: 

1. Abuja is the Federal Capital of Nigerian and Lagos is the largest commercial city of Nigeria 
2. Triangles have three sides implies that a triangle is a polygon. 
3. If a person is 15 years old, then the person is an adult.

1. Give the negation of the following 

1. An octagon has eight sides. 
2. The diagonals of an isosceles trapezium are equal 
3. 9 – 17 < 7 or 15 < (-6)2.

1. Using A and B, write down the inverse, converse and contrapositive of the following:

1. If Ibadan is the largest city in Nigeria, then it is the largest city in Oyo state. 
2. If a triangle has all its three sides equal, then it is an equilateral triangle 

1.  Draw a truth tables for the following 

(a) (P VQ)    (b) (P ⋀Q)        (c) (P Q) ⋀(P R) 

1. (a) copy and complete the table below: 

Where cond. = conditional, inv. = inverse, conv. = converse, contr. = contrapositive. 

(b) what do you notice about 

1. Converse and inverse statements? 
2. Conditional and contraposivite statement? 

1. All warm blooded animals are mammals.

Human beings are warm blooded animals

Therefore, human beings are mammals. 

1. All professors are meticulous. 

Salami is meticulous

Therefore Salami is a professor. 

GENERAL EVALUATION/REVISION QUESTIONS

1. Using truth tables, determine the validity of the following arguments: 
2. If I love my wife, then I will buy her a gift. 

I love my wife. 

Therefore, I will buy her a gift. 

2. All dogs can bark. 

This is not a dog. 

Therefore, it cannot bark.

3. If I am your friend, then I will drink alcohol. 

I do not drink alcohol.

Therefore, I am your friend. 

1. Using tables, determine whether or not the following arguments are valid. 
2. 2 + 5 = 9 or 3 + 4 < 2 + 1 

2 + 5 9 

Therefore, 3 + 4 ≮ 2 + 1 

5. 12of -12 = - 6 or 12 + 34= 54

12of -12 - 6

Therefore, 12+ 34 ≠ 54

6. If 2x + 5 = 15, then x = 5

x 5

Therefore, 2x + 5 15 

WEEKEND ASSIGNMENT

Objectives

1. The conditional statement P⇒Q is false when              A. both P and Q are true  B. P is true and Q is false  C. P is false and Q is true  D. P is false and Q is false. 
2. The negation of PʌQ is    A.  ~PʌQ  B. ~Pʌ~Q  C.  ~Pv~Q  D. ~(PvQ)

Given that p is the statement ‘Ayo has determination and q is the statement ‘Ayo willsucced’. Use the information to answer thesequestions.Which of these symbols represent these statements?

3.Ayo has no determination.A.    P ⇒ q      B.   ~p⇒ q          C.      ~p

4.If Ayo has no determination then he won’t succeed.A. ~p ⇒~ q B. p ⇒~ q C.p⇒ q

1.   p ⇒~ q
2. If Ayo wont succeed then he has no determination.A. ~q ⇒ p B. ~q ⇒~qC.~q ⇒p
3. q ⇒ p

Theory

• Using truth tables, determine the validity of the following arguments: 

1. When the weather is very hot you sweat profusely. 

When you sweat profusely your clothes get dirty. 

Therefore, when the weather is very hot your clothes get dirty. 

1. If it was an accident, something would have been broken 

Nothing was broken 

Therefore, it was not an accident. 

1. If you study mathematics, then you become an engineer. 

If you become an engineer, then you will be comfortable. 

Therefore, if you study mathematics then you will be comfortable.

1. The teacher is teaching maths or arts. 

The teacher is not teaching maths or arts.

Therefore, the teacher is teaching arts. 

• Using tables, determine whether or not the following arguments are valid. 

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

PQR does not have two equal sides. 

Therefore, PQR does not have two equal angles. 

1. If a triangle has two equal sides,

It is an isosceles

XYZ has two equal sides. 

Therefore, XYZ is an isosceles . 

3. Determine the validity of each of the following arguments. 

1. X is a square X is a rectangle. 

X is a square X is a rhombus. 

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

READING ASSIGNMENT

New General Mathematics SSS2, pages 218-223, exercise 20a and 20b.