Lesson Notes By Weeks and Term v4 - SHS 3

Lesson Video

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Performance objectives

• Define simple and compound statements.
• Use logical connectives like 'and', 'or', and 'if...then'.
• Construct compound statements from simple ones.
• Apply the chain rule of implication.
• Represent statement relationships with Venn diagrams.

Lesson summary

This lesson introduces the fundamental principles of mathematical logic, which is a powerful application of algebraic thinking. We will move beyond using letters to represent unknown numbers and start using them to represent entire ideas or statements. Understanding logic helps us to think clearly, construct valid arguments, and analyse information critically. In our daily lives in Ghana, from understanding news reports and political debates to using digital apps, logic is everywhere. This skill is crucial for success in higher education, computer science, law, and many other fields.

Lesson notes

This section breaks down the core ideas of mathematical logic needed for this lesson. 2.1 What is a Statement?

In mathematical logic, a statement (also called a proposition) is a declarative sentence that is either True or False, but not both. The truth value (True or False) of a statement is definite. Examples of Statements: "Accra is the capital city of Ghana." (This is True). "The River Volta is the longest river in Africa." (This is False). "5 + 7 = 12." (This is True). "A square has 3 sides." (This is False). Examples of NON-Statements: "What is your name?" (This is a question, not a statement). "Do your homework." (This is a command). "Jollof rice is delicious." (This is an opinion; it is not definitively true or false for everyone). "x + 3 = 5" (This is an open sentence. Its truth depends on the value of x. It's not a statement until x is defined). 2.2 Simple and Compound Statements Simple Statement: A statement that contains a single idea and cannot be broken down into simpler statements. We use lowercase letters like `p`, `q`, `r` to represent simple statements. `p`: It is raining today. `q`: The Black Stars won the match. `r`: 3 is an odd number. Compound Statement: A statement formed by combining two or more simple statements using connecting words called logical connectives. Example: "It is raining today and the Black Stars won the match." This compound statement is formed from the two simple statements `p` and `q` above. 2.3 Logical Connectives

These are the special words or symbols used to connect simple statements to form compound statements.

| Name of Connective | Connecting Word(s) | Symbol | How to Read | Example (p: He is tall, q: He is strong) | | :------------------ | :----------------- | :----: | :---------- | :--------------------------------------- | | Conjunction | `and` | `∧` | p and q | He is tall and he is strong. (`p ∧ q`) | | Disjunction | `or` | `∨` | p or q | He is tall or he is strong. (`p ∨ q`) | | Conditional | `if... then...` | `⇒` | if p then q | If he is tall, then he is strong. (`p ⇒ q`) | | Biconditional | `if and only if` | `⇔` | p iff q | He is tall if and only if he is strong. (`p ⇔ q`) |

Evaluation guide

• Identify and construct compound statements and form simple statements using the
• connectives.
• Talk for Learning, Group work.
• Compound and simple statements