Lesson Note
INTRODUCTION
Logical reasoning is a branch of mathematics that deals with the structure of arguments and the validity of conclusions drawn from premises. It helps students develop clear thinking, avoid fallacies, and solve problems that involve conditions, comparisons, and logical relationships. In everyday life, logical reasoning is used when deciding whether to take an umbrella based on the weather forecast, when planning a road trip to avoid traffic, or when judging the reliability of a news article. By mastering logical reasoning, students gain a powerful tool for critical thinking that applies to science, mathematics, and real‑world decision making.
MAIN CONTENT
Simple Statements and Truth Values
In logical reasoning, a simple statement (also called an atomic statement) is a declarative sentence that is either true or false. The truth value of a statement is denoted by T (true) or F (false). For example, “The sky is blue” is a true statement, whereas “The earth is flat” is false. The truth value depends on the actual state of the world.
Worked Example
Step 1: Identify the statement: “The number 7 is even.”
Step 2: Evaluate the statement. 7 divided by 2 gives 3 remainder 1, so it is not even.
Step 3: Assign truth value: F.
Answer: The statement “The number 7 is even” is false.
Practice Questions
Practice Question 1: Determine the truth value of the statement “12 is a multiple of 4.”
Answer: T
Practice Question 2: Evaluate the statement “The moon is made of cheese.”
Answer: F
Negation of Statements
Negation flips the truth value of a statement. If P is a statement, ¬P (read “not P”) is true when P is false, and false when P is true. The symbol for negation is ¬.
Worked Example
Step 1: Statement P: “The sum of 3 and 5 is 9.”
Step 2: Evaluate P. 3 + 5 = 8, so P is false (F).
Step 3: Negate P: ¬P is true (T).
Answer: ¬P is true.
Practice Questions
Practice Question 1: Negate the statement “The number 10 is prime.”
Answer: The statement “10 is not prime” is true.
Practice Question 2: Negate the statement “All cats are mammals.”
Answer: The statement “Not all cats are mammals” is false.
Conjunction (AND) and Disjunction (OR)
A conjunction (P ∧ Q) is true only when both P and Q are true. A disjunction (P ∨ Q) is true when at least one of P or Q is true. The symbols ∧ and ∨ represent “and” and “or” respectively.
Worked Example
Step 1: Let P: “2 + 2 = 4” (T).
Step 2: Let Q: “5 × 3 = 15” (T).
Step 3: Conjunction P ∧ Q: T ∧ T = T.
Step 4: Disjunction P ∨ Q: T ∨ T = T.
Answer: P ∧ Q is true; P ∨ Q is true.
Practice Questions
Practice Question 1: Evaluate the conjunction of “7 is odd” and “7 > 5”.
Answer: True ∧ True = True
Practice Question 2: Evaluate the disjunction of “4 is even” and “4 is prime”.
Answer: True ∨ False = True
Conditional Statements (If‑Then) and Contrapositive
A conditional statement has the form “If P, then Q” and is written P → Q. It is false only when P is true and Q is false; otherwise it is true. The contrapositive of P → Q is ¬Q → ¬P, which has the same truth value as the original statement.
Worked Example
Step 1: P: “The number 4 is even” (T).
Step 2: Q: “The number 4 is divisible by 2” (T).
Step 3: Conditional P → Q: T → T = T.
Step 4: Contrapositive ¬Q → ¬P: ¬T → ¬T = F → F = T.
Answer: Both the conditional and its contrapositive are true.
Practice Questions
Practice Question 1: Determine if “If a shape is a square, then it has four equal sides” is a true conditional.
Answer: True
Practice Question 2: Find the contrapositive of “If it rains, then the ground gets wet.”
Answer: “If the ground is not wet, then it does not rain.”
Biconditional Statements (If and Only If)
A biconditional statement P ↔ Q is true when P and Q have the same truth value (both true or both false). It is false when one is true and the other false. The symbol ↔ means “if and only if”.
Worked Example
Step 1: P: “The number 9 is a perfect square” (T).
Step 2: Q: “The square root of 9 is an integer” (T).
Step 3: Biconditional P ↔ Q: T ↔ T = T.
Answer: The biconditional statement is true.
Practice Questions
Practice Question 1: Evaluate “A number is even if and only if it is divisible by 2.”
Answer: True
Practice Question 2: Evaluate “A triangle is right‑angled if and only if it has three sides.”
Answer: False
De Morgan's Laws
De Morgan's laws provide rules for negating conjunctions and disjunctions:
¬(P ∧ Q) ≡ ¬P ∨ ¬Q
¬(P ∨ Q) ≡ ¬P ∧ ¬Q
These equivalences allow simplification of logical expressions.
Worked Example
Step 1: P: “The sky is clear” (T).
Step 2: Q: “It is raining” (F).
Step 3: Negate conjunction: ¬(P ∧ Q) = ¬(T ∧ F) = ¬F = T.
Step 4: Apply De Morgan: ¬P ∨ ¬Q = ¬T ∨ ¬F = F ∨ T = T.
Answer: Both sides evaluate to true, confirming the law.
Practice Questions
Practice Question 1: Use De Morgan's law to simplify ¬(A ∨ B).
Answer: ¬A ∧ ¬B
Practice Question 2: Simplify ¬(C ∧ D) using De Morgan's law.
Answer: ¬C ∨ ¬D
Logical Equivalence and Simplification
Two logical expressions are equivalent if they have the same truth value for all possible truth assignments. Common equivalences include:
P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)
(P ∧ Q) ∨ (P ∧ ¬Q) ≡ P
Simplifying expressions helps in solving logical problems efficiently.
Worked Example
Step 1: Expression: (P ∧ Q) ∨ (P ∧ ¬Q).
Step 2: Factor P: P ∧ (Q ∨ ¬Q).
Step 3: Recognize Q ∨ ¬Q is always true (T).
Step 4: Simplify: P ∧ T = P.
Answer: The expression simplifies to P.
Practice Questions
Practice Question 1: Simplify (R ∨ S) ∧ (R ∨ ¬S).
Answer: R
Practice Question 2: Show that ¬(¬P ∨ Q) ≡ P ∧ ¬Q.
Answer: True by De Morgan and double negation.
Inference and Syllogisms
Inference involves deriving a conclusion from given premises. A syllogism is a particular type of inference with two premises and one conclusion. The classic form is:
Premise 1: All A are B.
Premise 2: All C are A.
Conclusion: All C are B.
Worked Example
Step 1: Premise 1: “All mammals are warm‑blooded.”
Step 2: Premise 2: “All dogs are mammals.”
Step 3: Conclusion: “All dogs are warm‑blooded.”
Answer: The conclusion follows logically from the premises.
Practice Questions
Practice Question 1: Given “All fruits are edible” and “All apples are fruits,” conclude whether “All apples are edible.”
Answer: Yes
Practice Question 2: Given “No reptiles are mammals” and “All snakes are reptiles,” conclude whether “All snakes are mammals.”
Answer: No
SUMMARY
Logical reasoning equips students with the ability to analyze statements, combine them using logical operators, and draw valid conclusions. By mastering simple statements, negation, conjunction, disjunction, conditional and biconditional statements, De Morgan's laws, logical equivalence, and inference, learners can approach problems systematically and avoid errors. These skills are foundational not only for mathematics but also for everyday decision making, scientific reasoning, and critical evaluation of information.
EVALUATION
1. Evaluate the truth value of the statement: “If 2 + 2 = 5, then the moon is made of cheese.”
2. Determine the contrapositive of: “If a figure is a rectangle, then it has four right angles.”
3. Simplify the expression: ¬(P ∧ Q) ∨ (P ∧ ¬Q).
4. Using De Morgan’s laws, rewrite ¬(X ∨ Y) in its equivalent form.
5. From the premises “All birds can fly” and “All penguins are birds,” infer the conclusion about penguins.