SUBJECT: MATHEMATICS
CLASS: SS 2
DATE:
TERM: 2nd TERM
REFERENCE BOOKS
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
Logic Operator | Symbol |
And | â |
or | â |
if … then | |
if and only if | |
Not | ~ |
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
P | ~P |
T | F |
F | T |
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 | PâQ |
T | T | T |
T | F | F |
F | T | F |
F | F | F |
P | Q | PQâ |
T | T | T |
T | F | T |
F | T | T |
F | F | F |
PâQ is true when both PâQ is false when both P and Q are false.
P and Q are true
P | Q | PQ |
T | T | T |
T | F | F |
F | T | F |
F | F | T |
Figure 6
Figure 5
P | Q | PQ |
T | T | T |
T | F | F |
F | T | T |
F | F | T |
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.
Solution
P - -5 < 8 is true (T)
Q = 2 < -50 is false (F)
∴symbolic for: P â Q is false
(see 2nd row of fig 3)
Q (opposite angles of any quadrilateral are supplementary).
P is true (T) and Q is false (F)
∴ PâQ is true (see 2nd row of fig 4)
Q = a person is a teenager.
P is T and Q is F
∴ PQ is false (see 2nd row of fig 5)
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.
∴ PâºQ 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
(ii) If two triangles have their corresponding sides proportional, then they are equiangular.
(ii) If two triangles are not equiangular, then their corresponding sides are not proportional
(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 â∼Q) (P â∼Q)
Solution
Method 1
P | Q | ∼P | ∼Q | ∼P â ∼Q | P â∼Q | (P â∼Q) (P â∼Q) |
T | T | F | F | F | F | T |
T | F | F | T | T | T | T |
F | T | T | F | T | F | F |
F | F | T | T | T | F | F |
Explanation
Since there are two variables, P and Q, we will have 4 rows.
∼P â ∼Qis false when both P and ∼Q are false according to the table for V.
(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
P | Q | ∼P | ∼Q | (P â∼Q) (P â∼Q) | ||
T | T | F | F | F | T | F |
T | F | F | T | T | T | T |
F | T | T | F | T | F | F |
F | F | T | T | T | F | F |
(1) (2) (3) (4) (5)
Explanation
Enter P and Q columns as usual.
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 âQ) [(P) V (Q)] is a contradiction.
Solution
P | ~P | ~(~P) | Pâº~(~P) |
T | F | T | T |
F | T | F | T |
The truth table of P ~(~P) is always T, so it is a tautology
P | Q | (P â Q) [(~P) V(~Q)] | |||||
T | T | T | F | F | F | F | |
T | F | F | F | F | T | T | |
F | T | F | F | T | T | F | |
T | F | F | F | T | T | T |
(1) (5) (2) (4) (3)
Column (5) shows that the truth table of (P â 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 â~Q (b) ~(P â~Q) V~R
Solution
Substituting the truth values directly into the statement ~P â~Q, we have ~T â~T.
But ~T is the same as F.
∴ ~T â~T gives F â F
Simplify the disjunction: F
∴ The compound statement ~P â~Q is false.
Substituting the truth values: ~(T â~T) V~T
Within brackets, negate: ~(T â F) V~T
Simplify brackets: ~ F V~T
T V F
Simplify disjunction: T
∴ ~(P â~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 â 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.
iii. S3 is true, i.e. no one can be a teacher and a doctor. (T â 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:
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)
P | Q | P Q | (P Q) â p | [(P Q) â P] Q |
T | T | T | T | T |
T | F | F | F | T |
F | T | T | F | T |
F | F | T | F | T |
Since the compound statement
[(P Q) â 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
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
[(P Q) â (Q R)] (P R)
P Q R [(P Q) â (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) â (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
(a) (P VQ) (b) (P âQ) (c) (P Q) â(P R)
Cond. | Inv. | Conv | Contr. | ||||
P | Q | P | Q | P â¹Q | P ⇒∼ Q | Q P | Q ⇒∼P |
T | T | ||||||
T | F | ||||||
F | T | ||||||
F | T |
Where cond. = conditional, inv. = inverse, conv. = converse, contr. = contrapositive.
(b) what do you notice about
Human beings are warm blooded animals
Therefore, human beings are mammals.
Salami is meticulous
Therefore Salami is a professor.
GENERAL EVALUATION/REVISION QUESTIONS
I love my wife.
Therefore, I will buy her a gift.
This is not a dog.
Therefore, it cannot bark.
I do not drink alcohol.
Therefore, I am your friend.
2 + 5 9
Therefore, 3 + 4 â® 2 + 1
12of -12 - 6
Therefore, 12+ 34 ≠ 54
x 5
Therefore, 2x + 5 15
WEEKEND ASSIGNMENT
Objectives
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
Theory
When you sweat profusely your clothes get dirty.
Therefore, when the weather is very hot your clothes get dirty.
Nothing was broken
Therefore, it was not an accident.
If you become an engineer, then you will be comfortable.
Therefore, if you study mathematics then you will be comfortable.
The teacher is not teaching maths or arts.
Therefore, the teacher is teaching arts.
PQR does not have two equal sides.
Therefore, PQR does not have two equal angles.
It is an isosceles
XYZ has two equal sides.
Therefore, XYZ is an isosceles .
X is a square X is a rhombus.
Therefore, X is rectangle X is a rhombus.
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.
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