Logic, Statements and Truth Tables for SS 1

Logic, Statements and Truth Tables

Performance Objectives

By the end of the lesson, students should be able to:

1. explain and form simple statements;
2. describe and form compound statements;
3. distinguish between simple and compound statements;
4. identify logical operators and symbols;
5. determine the truth value of compound statements;
6. identify negations, conjunctions, disjunctions, conditional and bi-conditional statements.

Introduction to Logic

Logic is the study of reasoning. In mathematics, logic helps to analyse statements and arguments to determine if they are true or false. It provides a framework for clear and precise thinking, which is essential in problem-solving and understanding mathematical proofs.

Simple Statements

A statement is a declarative sentence that is either true or false, but not both. It expresses a complete thought and can be assigned a truth value.

The truth value of a statement is either True (T) or False (F).

For a sentence to be a statement in logic, it must be possible to objectively determine its truth or falsity. Questions, commands, and exclamations are not statements because they cannot be assigned a truth value.

Examples of simple statements include:

1. Lagos is the capital of Nigeria. (False)
2. 2 + 3 = 5. (True)
3. The sun rises in the east. (True)
4. Every square is a rectangle. (True)
5. All birds can fly. (False)

Examples of sentences that are NOT statements include:

1. What is your name? (A question)
2. Close the door. (A command)
3. Oh, what a beautiful day! (An exclamation)
4. x + 5 = 10. (Its truth value depends on the value of x, so it’s not definitively true or false without more information.)
5. He is a good student. (Subjective; “good” can mean different things to different people.)

Forming Simple Statements

Simple statements are basic sentences that convey a single fact or idea. They do not contain any logical connectives like “and”, “or”, “if…then”, or “not”.

To form a simple statement, ensure the sentence is declarative and its truth value can be determined. For example, from the situation “A student is studying”, a simple statement could be “The student is studying mathematics.”

Negation of Simple Statements

The negation of a statement is a new statement that has the opposite truth value of the original statement.

If a statement P is true, then its negation is false. If a statement P is false, then its negation is true.

The negation of a statement P is usually formed by adding “not” or “it is not true that” to the original statement. The symbol for negation is $\neg$. So, the negation of P is written as $\neg P$.

Examples of negations:

1. Let P be the statement: “Abuja is the capital of Nigeria.” (True)
   The negation $\neg P$ is: “Abuja is not the capital of Nigeria.” (False)
2. Let Q be the statement: “All cats can bark.” (False)
   The negation $\neg Q$ is: “Not all cats can bark.” or “It is not true that all cats can bark.” (True)
3. Let R be the statement: “The number 7 is an even number.” (False)
   The negation $\neg R$ is: “The number 7 is not an even number.” (True)

Compound Statements

A compound statement is a statement formed by combining two or more simple statements using logical connectives (also called logical operators).

The truth value of a compound statement depends on the truth values of its simple component statements and the type of logical connective used.

Types of Compound Statements

1. Conjunction: A compound statement formed by joining two simple statements with the word “and”.

   Example: “It is raining AND the sun is shining.”

2. Disjunction: A compound statement formed by joining two simple statements with the word “or”.

   Example: “I will eat rice OR I will eat beans.”

3. Implication (Conditional): A compound statement formed by joining two simple statements with the words “if…then”.

   Example: “IF it rains, THEN the ground will be wet.”

4. Bi-implication (Bi-conditional): A compound statement formed by joining two simple statements with the words “if and only if”.

   Example: “A triangle is equilateral IF AND ONLY IF all its sides are equal.”

Logical Operators and Symbols

Logical operators are words or symbols used to connect simple statements to form compound statements. Each operator has a specific symbol and meaning.

The main logical operators and their symbols are:

“, “IF…THEN”; “Bi-conditional”, “↔”, “IF AND ONLY IF”.
alt: Table of logical operators with their symbols and meanings.
caption: Common Logical Operators and their Symbols.
aspect_ratio: 16:9
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Let P and Q be simple statements. We can represent compound statements using these symbols:

1. Negation: $\neg P$ (read as “not P”)
2. Conjunction: $P \land Q$ (read as “P and Q”)
3. Disjunction: $P \lor Q$ (read as “P or Q”)
4. Conditional: $P \to Q$ (read as “If P, then Q”)
5. Bi-conditional: $P \leftrightarrow Q$ (read as “P if and only if Q”)

Truth Value of a Compound Statement (Truth Tables)

A truth table is a mathematical table used in logic to determine the truth value of a compound statement for all possible truth values of its simple component statements.

For a statement with $n$ simple components, there will be $2^n$ possible combinations of truth values, and thus $2^n$ rows in its truth table.

1. Truth Table for Negation ($\neg P$)

The negation of a statement P has the opposite truth value of P.

2. Truth Table for Conjunction ($P \land Q$)

A conjunction $P \land Q$ is true only when both P and Q are true. In all other cases, it is false.

3. Truth Table for Disjunction ($P \lor Q$)

A disjunction $P \lor Q$ is false only when both P and Q are false. In all other cases, it is true.

4. Truth Table for Conditional Statement ($P \to Q$)

A conditional statement $P \to Q$ is false only when P is true and Q is false. In all other cases, it is true.

This is often the most counter-intuitive truth table. Think of it as a promise: “If P happens, then Q will happen.” The promise is only broken (false) if P happens but Q does not.

5. Truth Table for Bi-conditional Statement ($P \leftrightarrow Q$)

A bi-conditional statement $P \leftrightarrow Q$ is true when P and Q have the same truth value (both true or both false). It is false when P and Q have different truth values.

Determining the Truth Value of Compound Statements

To determine the truth value of a compound statement, follow these steps:

1. Identify the simple statements involved and assign them variables (e.g., P, Q, R).
2. Determine the truth value of each simple statement.
3. Identify the logical operators connecting the statements.
4. Use the truth tables for each operator to find the truth value of the compound statement.

Worked Example 1: Conjunction

Determine the truth value of the statement: “Lagos is a state in Nigeria AND 2 + 2 = 5.”

Solution:

1. Let P be “Lagos is a state in Nigeria.”
   Let Q be “2 + 2 = 5.”
2. Determine truth values:
   P is True (T).
   Q is False (F).
3. The operator is “AND”, which is conjunction ($\land$). The compound statement is $P \land Q$.
4. Using the truth table for conjunction:
   When P is T and Q is F, $P \land Q$ is F.

Therefore, the statement “Lagos is a state in Nigeria AND 2 + 2 = 5” is False.

Worked Example 2: Disjunction

Determine the truth value of the statement: “The sun is cold OR a triangle has four sides.”

Solution:

1. Let P be “The sun is cold.”
   Let Q be “A triangle has four sides.”
2. Determine truth values:
   P is False (F).
   Q is False (F).
3. The operator is “OR”, which is disjunction ($\lor$). The compound statement is $P \lor Q$.
4. Using the truth table for disjunction:
   When P is F and Q is F, $P \lor Q$ is F.

Therefore, the statement “The sun is cold OR a triangle has four sides” is False.

Worked Example 3: Conditional

Determine the truth value of the statement: “IF 5 is an odd number, THEN 10 is an even number.”

Solution:

1. Let P be “5 is an odd number.”
   Let Q be “10 is an even number.”
2. Determine truth values:
   P is True (T).
   Q is True (T).
3. The operator is “IF…THEN”, which is conditional ($\to$). The compound statement is $P \to Q$.
4. Using the truth table for conditional:
   When P is T and Q is T, $P \to Q$ is T.

Therefore, the statement “IF 5 is an odd number, THEN 10 is an even number” is True.

Worked Example 4: Bi-conditional

Determine the truth value of the statement: “A number is divisible by 2 IF AND ONLY IF it is an even number.”

Solution:

1. Let P be “A number is divisible by 2.”
   Let Q be “It is an even number.”
2. Determine truth values:
   P is True (T) for any number that is even.
   Q is True (T) for any number that is even.
   (These two statements are logically equivalent, meaning they always have the same truth value.)
3. The operator is “IF AND ONLY IF”, which is bi-conditional ($\leftrightarrow$). The compound statement is $P \leftrightarrow Q$.
4. Using the truth table for bi-conditional:
   When P is T and Q is T, $P \leftrightarrow Q$ is T.

Therefore, the statement “A number is divisible by 2 IF AND ONLY IF it is an even number” is True.

Worked Example 5: Compound Statement with Negation

Determine the truth value of the statement: “It is NOT true that Abuja is in Ghana AND 3 + 4 = 7.”

Solution:

1. Let P be “Abuja is in Ghana.”
   Let Q be “3 + 4 = 7.”
2. Determine truth values:
   P is False (F).
   Q is True (T).
3. The compound statement involves negation and conjunction. It can be written as $\neg P \land Q$.
4. First, find the truth value of $\neg P$:
   Since P is F, $\neg P$ is T.
5. Now, find the truth value of $\neg P \land Q$:
   We have $\neg P$ as T and Q as T.
   Using the truth table for conjunction, when both components are T, the conjunction is T.

Therefore, the statement “It is NOT true that Abuja is in Ghana AND 3 + 4 = 7” is True.