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.
• Apply logical reasoning to analyze statements.

Lesson summary

In our daily lives in Ghana, we constantly use logic to make decisions. When a friend says, "If Hearts of Oak wins the match, then I will buy you waakye," you are dealing with a logical statement. In computing, law, and even in scientific arguments, we need a precise way to determine if complex statements are true or false. This branch of mathematics, called mathematical logic, uses algebraic symbols and methods to analyse statements. Today, we will learn how to break down these complex statements and test their truthfulness using a powerful tool called a truth table.

Lesson notes

This section breaks down the building blocks of mathematical logic. A. Simple Statements and Truth Values Statement (or Proposition): A declarative sentence that is either True (T) or False (F), but not both. The truthfulness (T) or falsity (F) of a statement is its truth value. Example 1: "Accra is the capital of Ghana." (This is a statement, and its truth value is T). Example 2: "The River Volta is longer than the River Nile." (This is a statement, and its truth value is F). Non-Example: "Come here!" (This is a command, not a statement). "What is your name?" (This is a question, not a statement). We use lowercase letters like `p`, `q`, and `r` to represent simple statements. `p`: It is raining. `q`: I will pass my WASSCE mathematics paper. B. Compound Statements and Logical Connectives A compound statement is formed by combining two or more simple statements using special words or symbols called logical connectives. We will study five key connectives. Negation (NOT) Symbol: ~ (called a tilde) Meaning: It reverses the truth value of a statement. It means "it is not the case that..." Example: Let `p`: "The Black Stars won the match." (Assume this is True) Then `~p`: "The Black Stars did not win the match." (This would be False) Truth Table for Negation: | `p` | `~p` | |:---:|:----:| | T | F | | F | T | Conjunction (AND) Symbol: ∧ Meaning: Connects two statements. The compound statement is True only if BOTH simple statements are true. Think of it as a strict condition. Example: Let `p`: "You studied for the test." Let `q`: "You attended all classes." `p ∧ q`: "You studied for the test AND you attended all classes." For this entire statement to be true, you must have done both things. If you did only one (or neither), the whole statement is false. Truth Table for Conjunction: | `p` | `q` | `p ∧ q` | |:---:|:---:|:-------:| | T | T | T | | T | F | F | | F | T | F | | F | F | F | Disjunction (OR) Symbol: ∨ Meaning: Connects two statements. The compound statement is True if AT LEAST ONE of the simple statements is true. This is an "inclusive OR". Example: Let `p`: "The school is offering scholarships to students from the Ashanti Region." Let `q`: "The school is offering scholarships to science students." `p ∨ q`: "The school is offering scholarships to students from the Ashanti Region OR to science students." A student who is a science student from the Volta Region gets the scholarship (q is true). A student who is an arts student from the Ashanti Region gets the scholarship (p is true). A science student from the Ashanti Region also gets the scholarship (both p and q are true). The only person who doesn't get it is an arts student from the Volta Region (both are false). Truth Table for Disjunction: | `p` | `q` | `p ∨ q` | |:---:|:---:|:-------:| | T | T | T | | T | F | T | | F | T | T | | F | F | F | Conditional (IF... THEN...) / Implication Symbol: → Meaning: Represents a promise or a hypothetical situation. `p → q` reads "if p, then q". `p` is the hypothesis (or antecedent) and `q` is the conclusion (or consequent). The statement is only False when the promise is broken: that is, when the hypothesis (`p`) is True, but the conclusion (`q`) is False. Example: Your father says, `p → q`: "IF you get an 'A' in Additional Maths (p), THEN I will buy you a new phone (q)." Case 1 (T → T): You get an 'A', and he buys you a phone. The promise was kept. (True) Case 2 (T → F): You get an 'A', but he does not buy you a phone. The promise was broken. (False) Case 3 (F → T): You don't get an 'A', but he buys you a phone anyway (maybe for your birthday). The promise was not broken. (True) Case 4 (F → F): You don't get an 'A', and he doesn't buy you a phone. The promise was not broken. (True) Truth Table for Conditional: | `p` | `q` | `p → q` | |:---:|:---:|:-------:| | T | T | T | | T | F | F | | F | T | T | | F | F | T | Bi-conditional (IF AND ONLY IF) Symbol: ↔ Meaning: Represents equivalence. `p ↔ q` means `p → q` AND `q → p`. The compound statement is True only when both simple statements have the SAME truth value (both true or both false). Example: Let `p`: "You are a student of this school." Let `q`: "You have the school's ID card." `p ↔ q`: "You are a student of this school IF AND ONLY IF you have the school's ID card." This means if you are a student, you must have the ID (T↔T). If you are not a student, you cannot have the ID (F↔F). It's false if one is true and the other is false (e.g., you are a student but have no ID). Truth Table for Bi-conditional: | `p` | `q` | `p ↔ q` | |:---:|:---:|:-------:| | T | T | T | | T | F | F | | F | T | F | | F | F | T |

Guided Practice (With Solutions)

Let's build truth tables for more complex statements step-by-step. The key is to add one column for each operation.

Question 1: Construct a truth table for the compound statement `~p ∨ q`.

Evaluation guide

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