Lesson Notes By Weeks and Term v3 - Senior Secondary 1
Logical reasoning
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Logical reasoning
Download the Lessonotes Mobile Nigeria 2025 app for faster lesson access on Android and iPhone.
Subject: Further Mathematics
Class: Senior Secondary 1
Term: 3rd Term
Week: 2
Theme: Pure Mathematics
This page supports the lesson note with a companion video and a short classroom-ready summary.
For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.
Recognize true and false statements Give examples of negation and contrapositive of statements Identify the antecedents of statements Identify and write the conditional of statements
This section provides a detailed explanation of the core concepts of logical reasoning relevant to the stated performance objectives. A. Statements A statement (or proposition) in logic is a declarative sentence that is either definitively true (T) or definitively false (F), but not both. It must have a truth value. Questions, commands, and exclamations are not statements because they cannot be assigned a truth value.
Characteristics of a Statement: It is a declarative sentence. It must be either true or false. It cannot be both true and false simultaneously.
Examples: 1. "Abuja is the capital of Nigeria." (This is a statement. Its truth value is TRUE.) 2. "The sun rises in the West." (This is a statement. Its truth value is FALSE.) 3. "2 + 3 = 5." (This is a statement. Its truth value is TRUE.) 4. "Go to the market." (This is a command, not a statement.) 5. "Is Professor Wole Soyinka a Nobel laureate?" (This is a question, not a statement.) 6. "Nigeria has 36 states and the Federal Capital Territory." (This is a statement. Its truth value is TRUE.) B. Negation of a Statement The negation of a statement is a new statement that has the opposite truth value of the original statement. If the original statement is true, its negation is false, and vice versa. The negation of a statement 'P' is denoted by '¬P' (read as "not P" or "It is not the case that P").
How to Form a Negation: Insert the words "It is not the case that..." before the statement. Insert the word "not" into the statement at an appropriate place. Replace words like "all" with "not all" or "some not"; "some" with "none" or "no"; "is" with "is not"; "every" with "not every".
Truth Table for Negation: | P | ¬P | | :---- | :---- | | True | False | | False | True |
Examples:
1. Statement (P): "Lagos is a state in Nigeria." (True) Negation (¬P): "Lagos is not a state in Nigeria." (False) OR "It is not the case that Lagos is a state in Nigeria."
2. Statement (Q): "All Nigerian cars are manufactured locally." (False) Negation (¬Q): "Not all Nigerian cars are manufactured locally." (True) OR "Some Nigerian cars are not manufactured locally."
3. Statement (R): "Some students are brilliant." (True/False depending on context) Negation (¬R): "No students are brilliant." OR "It is not the case that some students are brilliant." C. Conditional Statements (Implications) A conditional statement is a statement of the form "If P, then Q," where P and Q are themselves statements. P is called the antecedent (or hypothesis). It is the condition. Q is called the consequence (or conclusion). It is the result if the condition is met. The conditional statement "If P, then Q" is denoted by `P → Q`.
Truth Table for Conditional Statements: A conditional statement `P → Q` is only false when the antecedent (P) is true and the consequence (Q) is false. In all other cases, it is true. | P | Q | P → Q | | :---- | :---- | :---- | | True | True | True | | True | False | False | | False | True | True | | False | False | True | Explanation of Truth Table: T → T (True): "If I get a scholarship (T), then I will study abroad (T)." If both happen, the statement is true. T → F (False): "If I get a scholarship (T), then I will not study abroad (F)." If I get the scholarship but don't study abroad, the statement "If I get a scholarship, then I will study abroad" is false. F → T (True): "If I do not get a scholarship (F), then I will study abroad (T)." This doesn't contradict the original statement. Perhaps I found another way to study abroad. The original promise (conditional) is not broken. * F → F (True): "If I do not get a scholarship (F), then I will not study abroad (F)." scholarship (T), then I will not study abroad (F)." If I get the scholarship but don't study abroad, the statement "If I get a scholarship, then I will study abroad" is false. F → T (True): "If I do not get a scholarship (F), then I will study abroad (T)." This doesn't contradict the original statement. Perhaps I found another way to study abroad. The original promise (conditional) is not broken. F → F (True): "If I do not get a scholarship (F), then I will not study abroad (F)." This also doesn't contradict the original statement. The promise was about what would happen if I got a scholarship; since I didn't, the promise isn't violated.
Examples:
1. Statement: "If it rains heavily (P), then the roads will be flooded (Q)." Antecedent (P): "It rains heavily." Consequence (Q): "The roads will be flooded."
2. Statement: "If a number is even (P), then it is divisible by 2 (Q)." Antecedent (P): "A number is even." Consequence (Q): "It is divisible by 2."
3. Statement: "If you attend classes regularly (P), then you will understand the subject better (Q)." Antecedent (P): "You attend classes regularly." Consequence (Q): "You will understand the subject better." D. Contrapositive of a Conditional Statement For a conditional statement `P → Q` ("If P, then Q"), its contrapositive is `¬Q → ¬P` ("If not Q, then not P"). The contrapositive always has the same truth value as the original conditional statement. They are logically equivalent.
How to Form a Contrapositive:
1. Identify the antecedent (P) and the consequence (Q) of the conditional statement.
2. Negate both the antecedent and the consequence (`¬P` and `¬Q`).
3. Swap the positions: the negated consequence (`¬Q`) becomes the new antecedent, and the negated antecedent (`¬P`) becomes the new consequence. Truth Table for Contrapositive (and its Equivalence to Conditional): | P | Q | ¬P | ¬Q | P → Q | ¬Q → ¬P | | :---- | :---- | :---- | :---- | :---- | :------ | | True | True | False | False | True | True | | True | False | False | True | False | False | | False | True | True | False | True | True | | False | False | True | True | True | True | Note that P → Q and ¬Q → ¬P columns are identical.
Examples:
1. Conditional (P → Q): "If it rains heavily (P), then the roads will be flooded (Q)." Negation of P (¬P): "It does not rain heavily." Negation of Q (¬Q): "The roads will not be flooded." Contrapositive (¬Q → ¬P): "If the roads are not flooded, then it did not rain heavily." (Logically equivalent to the original statement.)
2. Conditional (P → Q): "If a student passes WAEC (P), then they can apply for university (Q)." Negation of P (¬P): "A student does not pass WAEC." Negation of Q (¬Q): "They cannot apply for university." Contrapositive (¬Q → ¬P): "If a student cannot apply for university, then they did not pass WAEC."
3. Conditional (P → Q): "If a number is divisible by 10 (P), then it is divisible by 5 (Q)." Negation of P (¬P): "A number is not divisible by 10." Negation of Q (¬Q): "It is not divisible by 5." * Contrapositive (¬Q → ¬P): "If a number is not divisible by 5, then it is not divisible by 10." Phase 1: Introduction (10 minutes)
Teacher Activity: Begins by writing various types of sentences on the board (e.g., "Nigeria is in West Africa.", "What time is it?", "Clean the board.", "This food is delicious."). Asks learners to identify which sentences can be judged as either true or false. Introduces the concept of a "statement" in logic, emphasizing its definitive truth value. Discusses why commands, questions, and exclamations are not logical statements.
Student Activity: Participate in identifying sentences that are statements. Discuss and give reasons for their choices. Provide simple examples of statements and non-statements.
Phase 2: Lesson Development - Part 1: Statements and Negation (20 minutes)
Teacher Activity: Formally defines a statement with examples relevant to Nigerian context (e.g., "The Naira is stronger than the US Dollar." - False; "Eko Bridge is in Lagos." - True). Explains the concept of truth value (True/False). Introduces the negation of a statement (¬P). Demonstrates how to form negations using different sentence structures and appropriate Nigerian examples (e.g., "All students are present." vs. "Not all students are present." or "Some students are absent."). Discusses the truth table for negation.
Student Activity: Identify truth values for given statements. Practice forming negations of simple statements provided by the teacher or peers. Engage in pair-work to generate statements and their negations.
Phase 3: Lesson Development - Part 2: Conditional Statements, Antecedent, Consequence, and Contrapositive (30 minutes)
Teacher Activity: Introduces conditional statements ("If P, then Q"). Clearly defines and identifies the antecedent (P) and consequence (Q) using several examples (e.g., "If you wake up early, then you will get to school on time."). Guides learners through the truth table for conditional statements, explaining each row with practical scenarios. Introduces the concept of the contrapositive (¬Q → ¬P) of a conditional statement. Demonstrates step-by-step how to form the contrapositive using examples, highlighting the negation and swapping of positions. Emphasizes the logical equivalence between a conditional statement and its contrapositive.
Student Activity: Identify antecedents and consequences in conditional statements. Construct conditional statements from two simple statements. Practice forming contrapositives of given conditional statements. Participate in group discussions to analyze the truth values of conditional statements based on various scenarios.
Phase 4: Guided Practice & Consolidation (15 minutes)
Teacher Activity: Provides 3-5 practice questions covering all objectives. Walks through the solutions with the class, explaining the reasoning behind each step. Answers questions and clarifies misconceptions.
Student Activity: Attempt practice questions individually or in small groups. Share answers and reasoning with the class. Ask questions for clarification.
Phase 5: Conclusion & Assignment (5 minutes)
Teacher Activity: Summarizes the key concepts covered: statements, negations, conditional statements, antecedents, consequences, and contrapositives. Assigns independent practice questions as homework.
Student Activity: Note down key points. Copy homework assignment.
Instructions: For each question, perform the indicated task.
Question 1: Consider the following sentences: a. "The Niger River flows through Nigeria." b. "How many states are there in Nigeria?" c. "2x + 1 = 7." (Assume x is an unknown variable) d. "Go to the polling unit." e. "Every Nigerian citizen has the right to vote." For each sentence, state whether it is a logical statement and, if so, determine its truth value.
Solution 1: a. "The Niger River flows through Nigeria." Is it a statement? Yes.
Truth Value: True. (The Niger River is a major river in West Africa, flowing through Nigeria.) b. "How many states are there in Nigeria?" Is it a statement? No.
Reason: It is a question and cannot be assigned a truth value. c. "2x + 1 = 7." Is it a statement? No.
Reason: This is an open sentence or an equation. Its truth value depends on the value of 'x'. Until 'x' is defined (e.g., x=3), it cannot be definitively true or false. d. "Go to the polling unit." Is it a statement? No.
Reason: It is a command or an imperative sentence. e. "Every Nigerian citizen has the right to vote." Is it a statement? Yes.
Truth Value: True. (This is a fundamental constitutional right in Nigeria for eligible citizens.)
Question 2: Write the negation of each of the following statements: a. "Kaduna is a city in Northern Nigeria." b. "All civil servants earn high salaries." c. "Some students passed the NECO examination." Solution 2: a.
Original Statement (P): "Kaduna is a city in Northern Nigeria." (True) Negation (¬P): "Kaduna is not a city in Northern Nigeria." (False) b.
Original Statement (P): "All civil servants earn high salaries." (False) Negation (¬P): "Not all civil servants earn high salaries." OR "Some civil servants do not earn high salaries." (True) c.
Original Statement (P): "Some students passed the NECO examination." (Potentially True) Negation (¬P): "No students passed the NECO examination." OR "It is not the case that some students passed the NECO examination." (Potentially False)
Question 3: For the following conditional statement, identify the antecedent and the consequence: "If a community has good healthcare facilities, then its residents will be healthier." Solution 3: Conditional Statement (P → Q): "If a community has good healthcare facilities, then its residents will be healthier." Antecedent (P): "A community has good healthcare facilities." Consequence (Q): "Its residents will be healthier." Question 4: Form a conditional statement from the following two simple statements: P: "The government invests in education." Q: "The nation will experience economic growth." Solution 4: Conditional Statement (P → Q): "If the government invests in education, then the nation will experience economic growth." Question 5: Write the contrapositive of the following conditional statement: "If you pay your electricity bill on time (P), then your power supply will not be disconnected (Q)." Solution 5: Original Conditional (P → Q): "If you pay your electricity bill on time (P), then your power supply will not be disconnected (Q)." Antecedent (P): "You pay your electricity bill on time." Consequence (Q): "Your power supply will not be disconnected." Negate P (¬P): "You do not pay your electricity bill on time." Negate Q (¬Q): "Your power supply will be disconnected." (
Note: The negation of "not be disconnected" is "be disconnected") Form Contrapositive (¬Q → ¬P): "If your power supply is disconnected, then you did not pay your electricity bill on time."
Analyzing Political and Media Claims: Logical reasoning helps Nigerian learners critically evaluate statements made by politicians, community leaders, or reported in the news. For example, analyzing statements like "If we vote for X, then our community will develop." Learners can identify the antecedent and consequence and assess the validity of such claims, considering whether the consequence truly follows from the antecedent, or if there are other factors involved. This promotes informed citizenship and reduces susceptibility to misinformation.
Everyday Decision Making and Planning: Nigerians constantly make decisions based on logical inferences.
For instance: "If I leave home by 6 AM (P), then I will avoid Lagos traffic (Q)." Or, "If I save a portion of my allowance (P), then I can buy a new textbook next month (Q)." Understanding conditional statements allows learners to plan effectively, anticipate outcomes, and identify potential flaws in their reasoning or others' advice.
Legal and Community Rules Interpretation: Many laws, regulations, and community rules are expressed in conditional forms. For example, "If you don't wear a face mask in public (P), then you will be fined (Q)." Or, "If you dispose of waste properly (P), then our community environment will be cleaner (Q)." Learning to identify antecedents and consequences, and understanding logical equivalence (like contrapositive), helps individuals interpret these rules correctly and understand their implications for behavior and compliance.