Further Mathematics - Senior Secondary 2 - Logical reasoning

Logical reasoning

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TERM: 1ST TERM

WEEK 11
Class: Senior Secondary School 2
Age: 16 years
Duration: 40 minutes of 4 periods
Subject: Further Mathematics
Topic: Logical Reasoning
Focus: Introduction to Propositional and Predicate Logical Resolution, Introduction to Theorem Proving.
SPECIFIC OBJECTIVES:
By the end of the lesson, students should be able to:

  1. Understand the concept of propositional logic and its applications.
  2. Explain predicate logic and how it differs from propositional logic.
  3. Demonstrate logical resolution for both propositional and predicate logic.
  4. Understand the process of theorem proving in logical reasoning.

INSTRUCTIONAL TECHNIQUES:

  • Lecture and demonstration
  • Question and answer
  • Discussion
  • Problem-solving exercises
  • Use of analogies and real-life applications

INSTRUCTIONAL MATERIALS:

  • Whiteboard and markers
  • Charts showing points to note in theorem proving
  • Example problems for propositional and predicate logical resolution
  • Worksheets for practice exercises

PERIOD 1 & 2: Introduction to Propositional and Predicate Logical Resolution

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces propositional logic, explaining its use of statements that are either true or false. Provides examples such as “The sky is blue.”

Students listen attentively and take notes. They ask clarifying questions about what makes a statement true or false.

Step 2 - Propositional Resolution

Demonstrates how propositional logic uses resolution to combine or simplify logical statements. Example: “If it rains, the ground will be wet” and “It is raining” leads to “The ground is wet.”

Students follow the teacher’s example and discuss real-life examples of propositional resolution.

Step 3 - Introduction to Predicate Logic

Introduces predicate logic, focusing on its use of variables and quantifiers (like ∀ and ∃). Example: “All humans are mortal” can be written as ∀x(Human(x) → Mortal(x)).

Students understand the difference between propositional and predicate logic. They engage in a discussion about the use of variables in logic.

Step 4 - Predicate Resolution

Demonstrates how to resolve statements in predicate logic. Example: “∀x (Human(x) → Mortal(x))” and “Socrates is a human” leads to “Socrates is mortal.”

Students work through the example with the teacher and discuss other examples.

NOTE ON BOARD:

  • Propositional Logic: Uses simple true/false statements.
  • Predicate Logic: Involves variables and quantifiers to express relationships.
  • Resolution: A method of combining or simplifying logical statements in both propositional and predicate logic.

EVALUATION (5 Exercises):

  1. Define propositional logic and give one example.
  2. What is a predicate in predicate logic?
  3. Convert the statement "All dogs are animals" into a predicate logic form.
  4. Demonstrate how propositional resolution works with two simple statements.
  5. What does the quantifier ∀ represent in predicate logic?

CLASSWORK (5 Questions):

  1. Write a simple propositional statement and identify if it is true or false.
  2. Write a statement in predicate logic using the quantifier ∀.
  3. Resolve the following logical statement: “If it is sunny, the ground will be dry” and “It is sunny.”
  4. Translate the statement “Some birds can fly” into predicate logic.
  5. Explain the meaning of the quantifier ∃ in predicate logic.

ASSIGNMENT (5 Tasks):

  1. Research and explain the difference between propositional and predicate logic.
  2. Demonstrate a real-life example of propositional logic in action.
  3. Provide a predicate logic example using the quantifier ∀.
  4. Convert the statement "Not all students pass the exam" into predicate logic.
  5. Solve a simple logical problem using resolution.

 

PERIOD 3 & 4: Introduction to Theorem Proving

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of theorem proving, explaining its importance in mathematics and logic.

Students listen and make notes on the importance of theorem proving in mathematics.

Step 2 - Proof by Contradiction

Demonstrates proof by contradiction. Example: To prove “√2 is irrational,” assume the opposite and show a contradiction.

Students follow along with the proof and discuss the process.

Step 3 - Direct Proof

Introduces direct proof, where we assume a hypothesis and show that it leads to the desired conclusion. Example: Proving “If n is even, then n² is even.”

Students work through the direct proof example with the teacher’s guidance.

Step 4 - Practice Problems

Provides the class with a series of simple theorems to prove using direct or contradiction methods.

Students work in pairs to prove theorems, asking for help as needed.

NOTE ON BOARD:

  • Proof by Contradiction: Assume the opposite of what you want to prove, and show that this leads to a contradiction.
  • Direct Proof: Start with a hypothesis and show the conclusion directly.
  • Theorems: Mathematical statements that can be proven true or false using logical reasoning.

EVALUATION (5 Exercises):

  1. Explain the method of proof by contradiction.
  2. Provide an example of a direct proof.
  3. Prove that the sum of two even numbers is even.
  4. Prove by contradiction that √3 is irrational.
  5. Demonstrate how theorem proving is used in mathematics.

CLASSWORK (5 Questions):

  1. Prove that if n is odd, then n² is odd.
  2. Prove by contradiction that “There is no largest prime number.”
  3. Write a simple direct proof of a mathematical statement.
  4. Provide an example where proof by contradiction is used.
  5. Solve a theorem-proving problem using direct proof.

ASSIGNMENT (5 Tasks):

  1. Research a famous theorem and summarize how it was proven.
  2. Prove by contradiction that “There are infinitely many prime numbers.”
  3. Write a direct proof of the statement: “If a number is divisible by 4, it is divisible by 2.”
  4. Solve a theorem-proving problem using both direct proof and proof by contradiction.
  5. Describe the importance of theorem proving in advanced mathematics.