Chapter Three: Logic – Additional Mathematics Form Two
Introduction
The term “Logic” comes from a Greek word “Logos” which means idea, thought, word, argument, account, reason, or principle. The interest in logic is not in the statements themselves, but in how the true and false statements are related to one another.
It is a reasonable way of thinking and understanding something. Formally, Logic is defined as the science or art that deals with principles of reasoning, and making appropriate decisions. In this chapter, you will learn about logical statements, truth tables, and arguments.
The competencies developed will help you to distinguish between valid and invalid arguments, reason logically and make proper decisions in daily life activities, construct circuit diagrams in the field of electronics, and make logical investigations on cases and judgements in the field of law, among many other applications.
Statements
A statement or proposition can be defined as a declarative sentence that can be either true or false but not both. The truth value of a statement is either true denoted by ‘T’ or false denoted by ‘F’.
Example 1
Determine whether or not each of the following sentences is a statement:
(a) Please mind your own business.
(b) x – y = z
(c) The sky is blue.
(d) Would you like some biscuits?
(e) Go to your class.
(f) One student is absent.
Solution
The sentences, (c) and (f) are statements (propositions) because their truth values are either true (T) or false (F). The sentences (a), (b), (d), and (e) are not statements (propositions) because they are neither true nor false. However, if the values of x, y and z are known, then (b) becomes a statement as it can be assigned a truth value.
Activity 1: Identifying statements from sentences
Individually or in a group perform the following tasks:
- Construct any eight sentences of your choice.
- From the sentences you have constructed in task 1, identify statements.
- What challenges have you faced in task 2?
- Share the results you have obtained in task 2 and 3 through discussions with other students for more inputs.
Types of Statements
There are two types of statements, namely simple and compound statements.
Simple Statement: Consists of a single declarative sentence that is either true or false.
Example: “Aisha likes singing”
Compound Statement: Formed by two or more declarative sentences.
Example: “Gabriella likes Mathematics but Musa likes History”
Truth Tables
A truth table is a table which indicates the truth value of a simple or compound statement containing several simple statements. The number of rows and columns of the truth table depends on the number of simple statements in a given compound statement.
Truth Table for One Proposition (p)
| p |
|---|
| T |
| F |
Truth Table for Two Propositions (p and q)
| p | q |
|---|---|
| T | T |
| T | F |
| F | T |
| F | F |
Negation of a Statement
Negation is the opposite of a given statement. The negation of a statement p is written as “~p” and it is read as “negation of p”.
Truth Table for ~p
| p | ~p |
|---|---|
| T | F |
| F | T |
Logical Connectives
Logical connectives are symbols used to join two or more propositions in a compound statement.
Conjunction (AND) – Symbol: ∧
The conjunction of two propositions, p and q is written as “p ∧ q” which is read as “p and q”. The truth value is true if and only if both p and q are true.
| p | q | p ∧ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Disjunction (OR) – Symbol: ∨
The disjunction of two propositions, p and q is written as “p ∨ q” and it is read as “p or q”. The truth value is false if and only if both propositions are false.
| p | q | p ∨ q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Conditional (Implication) – Symbol: →
The conditional statement “p → q” is read as “If p, then q” or “p implies q”. Its truth value is false when p is true and q is false.
| p | q | p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Biconditional (Double Implication) – Symbol: ↔
The biconditional statement “p ↔ q” is read as “p if and only if q”. The truth value is true if both p and q have the same truth value.
| p | q | p ↔ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | T |
Example 8
Write the statement “If she plays the drums, then he plays the trumpet” in symbolic form.
Solution
Let p: She plays the drums.
Let q: He plays the trumpet.
The statement becomes “if p, then q”, which is an implication statement.
Therefore, the symbolic form of the given statement is p → q.
Tautologies and Contradictions
A tautology is a compound statement which is always true for every truth value of its individual proposition.
Example: “Either Mwanaidi will go home or she will not go home.”
A contradiction is a compound statement which is always false for every truth value of its individual proposition.
Example: “5 is an even number and 5 is not an even number.”
Activity 3.2: Recognizing a tautology or contradiction
Individually or in a group, perform the following tasks:
- Construct at least four compound statements of your choice.
- Write the compound statements constructed in task 1 into symbolic form.
- Construct a truth table for each of the symbolic statements in task 2.
- Identify the truth value for each of the symbolic statement in task 3.
- Use the results in task 4 to recognize compounds statements which are tautology, contradiction, or neither.
- Share your results with other students through discussion for more inputs.
Arguments
An argument is a statement with a given set of hypotheses (premises) which yield a given conclusion.
Example
Test the validity of the argument “If I like music, then I will study. I did not study. Therefore, I don’t like music”.
Solution
Let p: I like music.
Let q: I will study.
The premises and conclusion of the argument are written as: p → q; ~q ∴ ~p
Thus, the argument in symbolic form is written as: [(p → q) ∧ ~q] → ~p
Using truth tables, we can verify that this argument is valid.
Activity 3: Recognizing validity of an argument
Individually or in a group, perform the following tasks:
- Construct at least six simple statements of your choice.
- From task 1, identify at least three related statements that form an argument.
- From task 2, write down the premises and the conclusion.
- Use the premises and the conclusion in task 3 to formulate the argument.
- Construct the truth table for the argument formulated in task 4.
- From task 5, identify the truth value of the argument.
- From task 6, with reason(s), state whether the argument is valid or not.
- Share your results with other students through discussion for more inputs.
Exercise 1
Determine whether each of the following statements is simple, compound or not a statement:
- A positive integer is a prime number only if it has no divisors other than 1 and itself.
- 3 < -5
- Go to sleep!
- If it rains tonight, then I will stay at home.
- What time is it?
Chapter Summary
- A statement or proposition can be defined as a declarative sentence that can be either true or false, but not both.
- There are two types of statements; namely simple and compound statements.
- The number of cases that describe the given compound statement depends on the number of simple statements in the compound statement.
- A truth table is a table that describes the truth values of a compound statement.
- Negation of a statement is written by introducing the word ‘not’ along with a verb.
- Logical connectives include conjunction (AND), disjunction (OR), conditional (implication), and biconditional (double implication).
- A tautology is a compound statement which is always true.
- A contradiction is a compound statement which is always false.
- An argument is a statement with a given set of hypotheses (premises) which yield a given conclusion.
- If an argument is a tautology, then the argument is valid, otherwise is not valid.
Revision Exercise
- Determine whether each of the following sentences is a simple statement, a compound statement, or not a proposition at all:
- What is your name?
- One year is equivalent to fifty weeks.
- A person is helpful if and only if he is kind.
- Let p be the statement “12 is divisible by 3” and q be the statement “2 is even”. Express each of the following proposition as an English sentence:
- ~q ∧ (q ∨ p)
- ~q ↔ (p ∧ q)
- Use truth tables to determine whether each of the following statements is a tautology, contradiction or neither:
- [(p → q) → q] → p
- ~[(~p ↔ q) ↔ (q ↔ p)]