Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Logical Reasoning
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Logical Reasoning
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Subject: General Mathematics
Class: Senior Secondary 1
Term: 1st Term
Week: 3
Theme: Algebraic Process
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Gives the meaning of simple statement and with examples Identify true or false statements State the negation of a simple statement Distinguish between simple statement and compound statement. Gives examples of conjunction,Disjunction,Implication and bi-implication. List the five logicaloperations and the irsymbols. Write the truthvalue of a compoundstatement in volvingany of the five logicaloperators.
2. 1. Statement A statement (or proposition) is a declarative sentence that is either true or false, but not both simultaneously. It is a sentence that can be assigned a truth value.
Examples: "Abuja is the capital of Nigeria." (True) "The sun rises in the West." (False) "Every Nigerian loves jollof rice." (This is subjective and cannot be definitively true or false for every Nigerian, thus it is NOT a statement in logic). "Go to the market." (An imperative sentence, NOT a statement). "What is your name?" (An interrogative sentence, NOT a statement). 2.
2. Simple Statement A simple statement is a statement that conveys a single idea and cannot be broken down into smaller statements. It is typically represented by a single letter, such as P, Q, R, etc.
Examples: P: "Lagos is a populated city." Q: "The Naira is Nigeria's currency." R: "2 + 3 = 5." 2.
3. Truth Value The truth value of a statement is either True (T or 1) or False (F or 0).
Examples: Statement: "Nigeria has 36 states." Truth Value: T Statement: "All birds can fly." Truth Value: F (e.g., Ostrich cannot fly) 2.
4. Negation (NOT) The negation of a statement P, denoted by `~P` (or `¬P`), is a statement that has the opposite truth value of P. If P is true, then `~P` is false, and vice versa. It is formed by adding "It is not true that..." or "It is false that..." before the statement, or by inserting "not" into the statement appropriately.
Symbol: `~` or `¬` Truth Table for Negation: | P | ~P | |---|----| | T | F | | F | T |
Examples: P: "Sokoto is a Northern Nigerian state." (T) ~P: "Sokoto is not a Northern Nigerian state." (F) Q: "All students are lazy." (F) ~Q: "It is not true that all students are lazy" OR "Some students are not lazy." (T) 2.
5. Compound Statement A compound statement is a statement formed by combining two or more simple statements using logical connectives (also known as logical operators).
Examples: "The sun is shining AND it is raining." "I will eat yam OR rice." "If you study hard, THEN you will pass." 2.
6. Logical Operations (Connectives) There are five fundamental logical operations that combine simple statements to form compound statements. Each operation has a specific symbol and truth table. 2.6.
1. Conjunction (AND) The conjunction of two statements P and Q, denoted by `P ∧ Q`, is true only if both P and Q are true. Otherwise, it is false.
Symbol: `∧` (read as "and")
Truth Table for Conjunction: | P | Q | P ∧ Q | |---|---|-------| | T | T | T | | T | F | F | | F | T | F | | F | F | F |
Examples: Let P: "Lagos is a state in Nigeria." (T)
Let Q: "Abuja is the capital of Nigeria." (T) P ∧ Q: "Lagos is a state in Nigeria AND Abuja is the capital of Nigeria." (T)
Let P: "Ayamase stew is red." (F, it's green)
Let Q: "Fufu is a swallow." (T) P ∧ Q: "Ayamase stew is red AND Fufu is a swallow." (F, because P is False) 2.6.
2. Disjunction (OR) The disjunction of two statements P and Q, denoted by `P ∨ Q`, is true if at least one of P or Q is true (i.e., P is true, or Q is true, or both are true). It is false only if both P and Q are false. This is typically the inclusive or.
Symbol: `∨` (read as "or")
Truth Table for Disjunction: | P | Q | P ∨ Q | |---|---|-------| | T | T | T | | T | F | T | | F | T | T | | F | F | F |
Examples: Let P: "Nigeria is in West Africa." (T)
Let Q: "Ghana is in true, or Q is true, or both are true). It is false only if both P and Q are false. This is typically the inclusive or.
Symbol: `∨` (read as "or")
Truth Table for Disjunction: | P | Q | P ∨ Q | |---|---|-------| | T | T | T | | T | F | T | | F | T | T | | F | F | F |
Examples: Let P: "Nigeria is in West Africa." (T)
Let Q: "Ghana is in East Africa." (F) P ∨ Q: "Nigeria is in West Africa OR Ghana is in East Africa." (T, because P is True)
Let P: "2 + 2 = 5." (F)
Let Q: "The moon is made of cheese." (F) P ∨ Q: "2 + 2 = 5 OR the moon is made of cheese." (F, because both P and Q are False) 2.6.
3. Implication / Conditional (IF...THEN...) The implication of P and Q, denoted by `P → Q`, means "If P, then Q." In this conditional statement, P is the antecedent (or hypothesis) and Q is the consequent (or conclusion). The statement `P → Q` is false only when the antecedent P is true, and the consequent Q is false. In all other cases, it is true.
Symbol: `→` (read as "if...then...")
Truth Table for Implication: | P | Q | P → Q | |---|---|-------| | T | T | T | | T | F | F | | F | T | T | | F | F | T |
Examples: P: "It rains." Q: "The ground gets wet." P → Q: "If it rains, then the ground gets wet." If (T) rain and (T) wet ground = (T) If (T) rain and (F) ground not wet (impossible) = (F) - This is the only false case. If (F) no rain and (T) wet ground (maybe from a tap) = (T) If (F) no rain and (F) ground not wet = (T) P: "You submit your assignment on time." Q: "You will get good grades." P → Q: "If you submit your assignment on time, then you will get good grades." (This statement is only false if you submit on time (T) but still get poor grades (F)). 2.6.
4. Bi-implication / Bi-conditional (IF AND ONLY IF) The bi-implication of P and Q, denoted by `P ↔ Q`, means "P if and only if Q." This statement is true when P and Q have the same truth value (i.e., both are true or both are false). It is false when P and Q have different truth values.
Symbol: `↔` (read as "if and only if" or "iff")
Truth Table for Bi-implication: | P | Q | P ↔ Q | |---|---|-------| | T | T | T | | T | F | F | | F | T | F | | F | F | T |
Examples: P: "A number is even." Q: "A number is divisible by 2." P ↔ Q: "A number is even IF AND ONLY IF it is divisible by 2." (T, as these statements are logically equivalent). P: "It is sunny." Q: "The temperature is high." * P ↔ Q: "It is sunny IF AND ONLY IF the temperature is high." (F, as it can be sunny but cold, or cloudy but hot). 2.6.
5. Summary of Logical Operations and Symbols: | Operation | Symbol | Read As | |-------------------|--------|------------------| | Negation | `~` or `¬` | NOT | | Conjunction | `∧` | AND | | Disjunction | `∨` | OR | | Implication | `→` | IF...THEN... | | Bi-implication | `↔` | IF AND ONLY IF | 3.
1. Introduction (10 minutes)
Teacher Activity: Begin the lesson by writing several sentences on the board. "Nigeria is a country in Africa." "How are you today?" "Wash your hands." "This chalk is green." (Assume it's white for a clear false statement) "x + 5 = 10." (
Note: This is an open sentence, not a statement until x is defined).
Student Activity: Students identify which sentences can be judged as true or false. Discuss why others cannot. This leads to the definition of a statement. 3.
2. Simple Statements and Truth Values (15 minutes)
Teacher Activity: Explain the formal definition of a statement and a simple statement. Provide more examples and ask students to determine their truth values. Emphasize that statements must be declarative and unequivocally true or false. Use relevant Nigerian examples. "Lagos is the capital of Nigeria." (F) "The President of Nigeria lives in Aso Rock." (T) "All Nigerian rivers flow from North to South." (F)
Student Activity: Students individually write down 3 simple statements about their environment or Nigeria and assign a truth value (T/F) to each. They then share with a partner. 3.
3. Negation of Statements (10 minutes)
Teacher Activity: Explain the concept of negation and its symbol (`~`). Demonstrate how to form the negation of a statement and how it reverses the truth value.
Student Activity: Provide students with 3-4 simple statements. Students write the negation for each and state its truth value. P: "All Nigerians are farmers." (F) ~P: "Not all Nigerians are farmers." (T) 3.
4. Compound Statements and Introduction to Logical Connectives (25 minutes)
Teacher Activity: Introduce compound statements as combinations of simple statements. Introduce the concept of logical connectives. Begin with Conjunction (AND, ∧): Explain its meaning and construct its truth table step-by-step. Use clear examples. "The traffic is heavy AND the road is bad." Introduce Disjunction (OR, ∨): Explain its meaning and construct its truth table. Emphasize the inclusive 'or'. "I will eat rice OR I will eat beans." Introduce Implication (IF...THEN..., →): Explain its meaning. Pay special attention to the case where P is True and Q is False (this is the only case where P → Q is False). Provide intuitive examples. "If you work hard, then you will succeed." Introduce Bi-implication (IF AND ONLY IF, ↔): Explain its meaning (equivalence). Construct its truth table. "A polygon is a triangle IF AND ONLY IF it has three sides." Student Activity: Group Work: Divide students into small groups. Each group is given a set of simple statements (e.g., P: "The sun is shining," Q: "It is hot"). Groups are to form compound statements using each of the connectives (∧, ∨, →, ↔) and discuss their expected truth values based on real-life conditions. Students practice constructing basic truth tables for 2 simple statements and one connective. 3.
5. Consolidation and Practice (10 minutes)
Teacher Activity: Review all five logical operations and their symbols. Facilitate a quick Q&A session to clear any misconceptions. Present a compound statement and guide students to construct its truth table.
Student Activity: Students work through a guided practice example of constructing a truth table for a slightly more complex compound statement (e.g., `~P ∨ Q`).
Question 1: For each of the following sentences, state whether it is a simple statement, a compound statement, or neither. If it is a statement, state its truth value (True or False). a) "Lagos is the most populous city in Nigeria." b) "Go and fetch water!" c) "If you invest in agriculture, then you contribute to national food security." d) "Are you from Kano State?" Solution 1: a) Simple Statement.
Truth Value: True. b) Neither. It is an imperative sentence (a command). c) Compound Statement. It is an implication. d) Neither. It is an interrogative sentence (a question).
Question 2: Write the negation of each of the following statements and determine the truth value of the negation. a) P: "All Nigerian politicians are corrupt." b) Q: "The square root of 9 is 3." c) R: "Some students are brilliant." Solution 2: a) ~P: "Not all Nigerian politicians are corrupt," or "Some Nigerian politicians are not corrupt." Truth Value of P: False (It is a generalization, not all are corrupt). Truth Value of ~P: True. b) ~Q: "The square root of 9 is not 3." Truth Value of Q: True. Truth Value of ~Q: False. c) ~R: "No students are brilliant," or "All students are not brilliant." Truth Value of R: True (Assuming there's at least one brilliant student). Truth Value of ~R: False.
Question 3: Identify the logical connective used in each compound statement below and rewrite the statement using logical symbols (P, Q, and the appropriate connective). a) "The rainy season has started AND farmers are planting crops." b) "I will eat cassava OR I will eat maize." c) "If a number is even, then it is divisible by 2." d) "You can vote IF AND ONLY IF you are 18 years old and registered." Solution 3: a)
Connective: Conjunction (AND)
Let P: "The rainy season has started." Let Q: "Farmers are planting crops." Symbolic form: `P ∧ Q` b)
Connective: Disjunction (OR)
Let P: "I will eat cassava." Let Q: "I will eat maize." Symbolic form: `P ∨ Q` c)
Connective: Implication (IF...THEN...)
Let P: "A number is even." Let Q: "It is divisible by 2." Symbolic form: `P → Q` d)
Connective: Bi-implication (IF AND ONLY IF)
Let P: "You can vote." Let Q: "You are 18 years old and registered." Symbolic form: `P ↔ Q` Question 4: Construct a truth table for the compound statement `P ∧ (~Q)`.
Solution 4: | P | Q | ~Q | P ∧ (~Q) | |---|---|----|----------| | T | T | F | F | | T | F | T | T | | F | T | F | F | | F | F | T | F |
Commentary: First, determine the truth values for `~Q` by negating the truth values of Q. Then, apply the conjunction rule (AND) between P and the calculated `~Q` column. Remember, for conjunction, the result is only true if both components are true.
Question 5: Construct a truth table for the compound statement `(P ∨ Q) → Q`.
Solution 5: | P | Q | P ∨ Q | (P ∨ Q) → Q | |---|---|-------|-------------| | T | T | T | T | | T | F | T | F | | F | T | T | T | | F | F | F | T |
Commentary: First, determine the truth values for `P ∨ Q`. Then, apply the implication rule (`→`) using `P ∨ Q` as the antecedent and `Q` as the consequent. Remember, implication is only false when the antecedent is true and the consequent is false.
This occurs in the second row: `P ∨ Q` is T, but `Q` is F, so the implication is F.
Consumer Decisions and Advertisement Analysis: Students can apply logical reasoning to analyze advertisements or product claims. For example, an ad might say, "If you use Brand X soap, then you will have clear skin." Students can identify this as an implication and evaluate its truth based on personal experience or critical thinking, understanding that correlation does not always imply causation. This helps them become more discerning consumers in Nigerian markets. Community Development and Policy Evaluation: Logical reasoning is vital in evaluating community projects or government policies. For instance, a local government might propose, "If we fix this road, then economic activity in the area will increase." Students can dissect such statements, considering the logical flow and potential hidden assumptions, to assess the validity of the proposed outcome. This fosters civic engagement and critical evaluation of leadership in their local communities. Digital Literacy and Information Verification: In an age of widespread misinformation and fake news, particularly prevalent on social media platforms in Nigeria, logical reasoning helps students evaluate the credibility of information. By breaking down complex arguments or claims into simple statements and assessing the logical connections (or lack thereof), students can identify logical fallacies and avoid being misled. For example, "This news is true because many people are sharing it" is a logical fallacy (appeal to popularity).