Lesson Notes By Weeks and Term v4 - SHS 3

Lesson Video

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Performance objectives

• Define conjunction, disjunction, implication, and negation.
• Construct truth tables for compound statements.
• Evaluate the truth values of compound statements.
• Formulate compound statements from simple ones.

Lesson summary

This lesson introduces the fundamentals of mathematical logic, a powerful tool used in reasoning and problem-solving. We will explore how simple ideas or statements can be combined to form more complex ones. In our daily lives in Ghana, we constantly make decisions based on conditions: "If the tro-tro is full, then I will wait for the next one," or "I will buy either waakye or kenkey for lunch." This lesson provides the mathematical structure to analyse such statements precisely. Understanding logic is crucial not only for mathematics but also for fields like computer programming (which powers our phones and apps), law, and even for constructing clear and persuasive arguments in debates.

Lesson notes

This topic deals with propositional logic, which is the study of statements and how they can be connected. 2.1. Statements (Propositions) A statement (or proposition) is a declarative sentence that is either True (T) or False (F), but not both. The truth value of a statement is its classification as true or false. Examples of statements: `p`: Accra is the capital of Ghana. (True) `q`: The Volta River is the longest river in Africa. (False) `r`: 5 + 7 = 12. (True) Examples of non-statements: "What is your name?" (This is a question) "Finish your homework." (This is a command) "x + 3 = 5" (This is an open sentence; its truth depends on the value of x) 2.2. Simple vs. Compound Statements A simple statement is a statement that contains a single idea. The examples `p`, `q`, and `r` above are simple statements. A compound statement is formed by combining two or more simple statements using words called logical connectives. 2.3. Logical Connectives and their Truth Tables

We use symbols to represent statements (e.g., p, q, r) and connectives. Let's explore the main connectives. Negation (NOT) Symbol: `~` (read as "not") Function: It reverses the truth value of a statement. Ghanaian Context: If `p` is "The Black Stars won the match," then `~p` is "The Black Stars did not win the match." Truth Table for Negation: | `p` | `~p` | |:---:|:----:| | T | F | | F | T | Conjunction (AND) Symbol: `∧` (read as "and") Function: It connects two statements. The compound statement `p ∧ q` is true only if both `p` and `q` are true. Ghanaian Context: "You need a passport (`p`) and a visa (`q`) to travel to the UK." You must have both; one is not enough. Truth Table for Conjunction: | `p` | `q` | `p ∧ q` | |:---:|:---:|:-------:| | T | T | T | | T | F | F | | F | T | F | | F | F | F | Disjunction (OR) Symbol: `∨` (read as "or") Function: It connects two statements. The compound statement `p ∨ q` is true if at least one of the statements (`p` or `q`) is true. It is only false when both are false. Ghanaian Context: "To gain admission, you need a credit in Mathematics (`p`) or a credit in Integrated Science (`q`)." You are fine if you have one, or the other, or both. Truth Table for Disjunction: | `p` | `q` | `p ∨ q` | |:---:|:---:|:-------:| | T | T | T | | T | F | T | | F | T | T | | F | F | F | Implication / Conditional (IF...THEN...) Symbol: `→` (read as "implies" or "if...then") Function: The statement `p → q` suggests a condition. `p` is called the antecedent (hypothesis) and `q` is the consequent (conclusion). The implication `p → q` is false only when a true antecedent leads to a false consequent. In all other cases, it is true. Ghanaian Context: A parent promises, "If you pass your WASSCE (`p`), then I will buy you a new phone (`q`)." You pass (T) and they buy the phone (T) -> Promise kept (T). You pass (T) and they don't buy the phone (F) -> Promise broken (F). You don't pass (F) but they still buy the phone (T) -> Promise not broken (T). You don't pass (F) and they don't buy the phone (F) -> Promise not broken (T). Truth Table for Implication: | `p` | `q` | `p → q` | |:---:|:---:|:-------:| | T | T | T | | T | F | F | | F | T | T | | F | F | T | 2.4. Constructing Truth Tables for Complex Statements

To construct a truth table for a more complex compound statement, we follow a systematic process.

Step-by-Step Procedure: Count the Variables: Identify the number of distinct simple statements (e.g., p, q, r). Let this number be `n`. Determine the Number of Rows: The truth table will have `2^n` rows of truth values. (2 variables = 2² = 4 rows; 3 variables = 2³ = 8 rows). Set up Columns: Create a column for each simple statement. Then, create columns for any smaller components of the compound statement, building up to the final statement. Fill Initial Columns: Fill the truth values for the simple statements systematically. For 2 variables (p, q): Column `p`: T, T, F, F Column `q`: T, F, T, F For 3 variables (p, q, r): Column `p`: T, T, T, T, F, F, F, F Column `q`: T, T, F, F, T, T, F, F Column `r`: T, F, T, F, T, F, T, F Evaluate Step-by-Step: Use the basic truth tables for the connectives to fill in the remaining columns, working from the inside out (just like with brackets in algebra).

Evaluation guide

• Use conjunction, disjunctions, implications and negations to construct truth tables of
• compound statements.
• Talk for Learning, and Group work.
• statements.