Logic – Statements, Implication, Converse, Equivalence, Negation, and Valid Argument

Download the Lessonotes Mobile Liberia app for faster lesson access on Android and iPhone.

Subject: Mathematics

Semester: 2

Period: 4

Week: 23

---

WEEK 23

Class: Grade 12
Age: 17 years
Duration: 40 minutes × 5 periods
Subject: Mathematics
Topic: Logic – Statements, Implication, Converse, Equivalence, Negation, and Valid Argument

FOCUS:

Introduction to mathematical logic involving statements, implications, converses, equivalence, negations, and determining the validity of arguments using logical reasoning and Venn diagrams.

SPECIFIC OBJECTIVES:

By the end of the lesson, students should be able to:

1. Identify and form true or false statements.
2. Form the negation of simple statements.
3. Draw conclusions using the implication sign (→).
4. Deduce an equivalent implication from a given implication.
5. Use Venn diagrams to determine the validity or otherwise of an implication or conclusion.

 

INSTRUCTIONAL TECHNIQUES:

• Question and answer
• Guided discovery
• Discussion
• Practice exercises
• Use of Venn diagrams

 

INSTRUCTIONAL MATERIALS:

• Charts showing logical symbols (¬, ∧, ∨, →, ↔)
• Flashcards with true/false statements
• Venn diagram charts
• Whiteboard and markers

 

PERIOD 1 & 2: Statements and Negation

PRESENTATION

Step	Teacher’s Activity	Student’s Activity
Step 1 – Introduction	Introduces the idea of statements: sentences that are either true or false (not both). Examples: “2 + 3 = 5” (true), “7 is an even number” (false).	Students listen, ask questions, and suggest their own examples.
Step 2 – Negation	Explains negation using the symbol (¬). Example: p: “It is raining.” ¬p: “It is not raining.”	Students rewrite statements in their negated forms.
Step 3 – Practice	Gives several statements for students to negate.	Students attempt exercises and present answers.

NOTE ON BOARD:

• A statement is a sentence that can be either true (T) or false (F).
• Negation: if p is a statement, ¬p means “not p.”

EVALUATION (5 Exercises):

1. State whether “5 + 7 = 12” is true or false.
2. Negate: “All triangles have three sides.”
3. Negate: “10 is greater than 20.”
4. Decide if “Lagos is in Nigeria” is true or false.
5. Negate: “Every student is hardworking.”

CLASSWORK (5 Questions):

1. Identify whether “2 is an odd number” is true or false.
2. Negate: “The sun rises in the east.”
3. Negate: “A square has four sides.”
4. State the truth value of “12 is divisible by 3.”
5. Negate: “All teachers are female.”

ASSIGNMENT (5 Tasks):

1. Write two true and two false mathematical statements.
2. Negate: “All prime numbers are odd.”
3. Negate: “Some cars are red.”
4. State whether “0 is a natural number” is true or false.
5. Negate: “Every real number is positive.”

 

PERIOD 3: Implication and Converse

PRESENTATION

Step	Teacher’s Activity	Student’s Activity
Step 1 – Introduction	Explains implication: If p then q (p → q). Example: If it rains (p), then the ground is wet (q).	Students listen and give similar real-life examples.
Step 2 – Converse	Explains that the converse of “if p then q” is “if q then p.” Example: If it rains then the ground is wet → Converse: If the ground is wet, then it rained.	Students try forming converses of given implications.
Step 3 – Practice	Teacher provides more examples.	Students construct converses.

NOTE ON BOARD:

• Implication: p → q
• Converse: q → p

EVALUATION (5 Exercises):

1. Write the implication: “If a number is even, then it is divisible by 2.”
2. State the converse of question 1.
3. State the converse of: “If a figure is a square, then it has four equal sides.”
4. Form an implication: p = “x > 2”, q = “x² > 4”.
5. Write the converse of question 4.

CLASSWORK (5 Questions):

1. If p: “A student studies,” q: “He passes,” write p → q.
2. Find the converse of “If a number is divisible by 6, then it is divisible by 2.”
3. Write implication: “If a shape is a rectangle, then it has four sides.”
4. Form converse of: “If a number ends in 0, then it is divisible by 5.”
5. Write implication: “If today is Monday, then tomorrow is Tuesday.”

ASSIGNMENT (5 Tasks):

1. State implication: “If a number is odd, then it is not divisible by 2.”
2. Find the converse of: “If it is a bird, then it has wings.”
3. Write implication: “If an angle is 90°, then it is a right angle.”
4. Form converse of: “If x = 2, then x² = 4.”
5. Write implication: “If it is sunny, then it is daytime.”

 

PERIOD 4: Equivalence and Valid Arguments

PRESENTATION

Step	Teacher’s Activity	Student’s Activity
Step 1 – Equivalence	Explains that an implication and its converse may both be true, then they are equivalent: p ↔ q. Example: “x is even ↔ x is divisible by 2.”	Students provide examples of equivalence.
Step 2 – Valid Argument	Introduces valid arguments and truth tables, showing how to test for validity. Uses Venn diagrams for visual explanation.	Students listen and practice with truth tables and diagrams.
Step 3 – Practice	Works out examples with class.	Students practice and solve in pairs.

NOTE ON BOARD:

• Equivalence: p ↔ q (p if and only if q)
• Valid argument: an argument is valid if the conclusion follows logically from the premises.

EVALUATION (5 Exercises):

1. State if “x is an even number ↔ x is divisible by 2” is true.
2. Write the equivalence of: “A number is a multiple of 3 ↔ it is divisible by 3.”
3. Use a truth table to check: p → q, p, therefore q.
4. State whether “If today is Sunday, tomorrow is Monday ↔ If tomorrow is Monday, today is Sunday” is equivalent.
5. Test if the conclusion is valid: All boys are tall; John is a boy; therefore John is tall.

CLASSWORK (5 Questions):

1. Write an equivalence involving odd numbers.
2. State whether “p ↔ ¬¬p” is valid.
3. Write the equivalence of “If a polygon has 3 sides, then it is a triangle.”
4. Use Venn diagram to check validity of: All dogs are animals. Bingo is a dog. Therefore Bingo is an animal.
5. State whether “x² = 9 ↔ x = 3” is correct.

ASSIGNMENT (5 Tasks):

1. Write equivalence for: “A square is a rectangle ↔ A rectangle with equal sides is a square.”
2. Use truth table to verify: (p → q) ↔ (¬q → ¬p).
3. Use Venn diagram to test: All humans are mortal; Socrates is human; therefore Socrates is mortal.
4. State whether “A number is divisible by 4 ↔ It is even and divisible by 2” is true.
5. Write equivalence for: “If an angle is 180°, then it is a straight angle.”

 

PERIOD 5: Consolidation & General Practice

• Teacher revises all subtopics: Statements, Negation, Implication, Converse, Equivalence, and Valid Argument.
• Students solve mixed problems combining all.
• Peer discussion and correction.