Mathematical Reasoning Explained: Logic, Statements, and Proofs
PUC Mathematics | Mathematical Reasoning | Daniel Classes
Why This Chapter Feels Different
Every math chapter you’ve taken has asked: solve this, calculate that, find x. Mathematical Reasoning asks something different: is this statement true? Can you prove it?
This is where pure logic lives. No formulas. Just clear thinking about what’s true, what’s false, and how to connect statements into valid arguments.
Most students avoid this chapter or memorize the truth tables without understanding why they matter. The chapter gets scored as “easy” because it’s just definitions. But PUC exams use it to test whether you can actually *reason*, not just calculate.
Here’s the core insight: Mathematical reasoning is about building chains of true statements that lead to conclusions. If you can do that, you’ve mastered the chapter.
Statements and Truth Values
A statement is a sentence that is either true or false (not both, and not neither).
Examples:
- “2 + 2 = 4” (true statement)
- “3 > 5” (false statement)
- “What time is it?” (not a statement — it’s a question)
- “x = 5” (not a statement in standard logic — it depends on x)
Truth value: Each statement has a truth value — either True (T) or False (F).
Logical Operators
Negation (~)
Definition: ~p is true if p is false, and false if p is true.
Example: If p = “It is raining,” then ~p = “It is not raining.”
Conjunction (∧)
Definition: p ∧ q is true only if both p and q are true.
Example: p = “I am happy,” q = “I am healthy.” p ∧ q = “I am happy AND healthy.”
Disjunction (∨)
Definition: p ∨ q is true if at least one of p or q is true.
Example: p = “I like coffee,” q = “I like tea.” p ∨ q = “I like coffee OR tea (or both).”
Implication (→)
Definition: p → q is false only when p is true and q is false.
Example: p = “It rains,” q = “The ground is wet.” p → q = “If it rains, then the ground is wet.”
Important: If p is false, then p → q is true (regardless of q). This confuses many students.
Biconditional (↔)
Definition: p ↔ q is true if p and q have the same truth value.
Example: “A triangle is equilateral if and only if all its sides are equal.”
Truth Tables
Conjunction (∧):
| p | q | p ∧ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Disjunction (∨):
| p | q | p ∨ q |
|---|---|---|
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
Implication (→):
| p | q | p → q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Tautologies, Contradictions, and Contingencies
Tautology: A statement that’s always true, regardless of the truth values of its parts.
Example: p ∨ ~p (a statement or its negation is always true)
Contradiction: A statement that’s always false.
Example: p ∧ ~p (a statement and its negation can’t both be true)
Contingency: A statement that’s sometimes true and sometimes false.
Example: p → q (depends on the truth values of p and q)
De Morgan’s Laws
Law 1: ~(p ∧ q) ≡ ~p ∨ ~q
“Not (both p and q)” is the same as “(not p) or (not q)”
Law 2: ~(p ∨ q) ≡ ~p ∧ ~q
“Not (p or q)” is the same as “(not p) and (not q)”
Example: “It’s not true that (Alice is smart AND Bob is tall)” means “(Alice is not smart) OR (Bob is not tall).”
How PUC Exams Test Reasoning
Type 1: Truth Tables (2-3 marks)
Example: “Construct a truth table for (p → q) ∧ r.”
Solution: Build all 8 rows (since there are 3 variables), compute the intermediate and final columns.
Type 2: Validating Statements (3-5 marks)
Example: “Is p ∨ ~p a tautology?”
Solution: Build the truth table. Check if it’s always true. (Yes, it is.)
Type 3: Logical Equivalence (3-5 marks)
Example: “Show that ~(p ∧ q) ≡ ~p ∨ ~q (De Morgan’s Law).”
Solution: Build truth tables for both sides and verify they’re identical.
The Five Common Mistakes
Mistake 1: Confusing Implication Direction
Example: “If p → q is true, is q → p also true?”
Wrong: “Yes, because they’re related.”
Right: “Not necessarily. p → q being true doesn’t tell us about q → p.”
Fix: Implication is one-directional. p → q does NOT mean q → p.
Mistake 2: Thinking p → q Is False When p Is False
Example: “Evaluate ‘If 2 > 3, then 5 = 5’.”
Wrong: “It’s false because 2 is not greater than 3.”
Right: “It’s true. When the hypothesis (p) is false, the implication is true regardless of the conclusion.”
Fix: p → q is false ONLY when p is true and q is false.
Mistake 3: Misapplying De Morgan’s Laws
Example: “Negate: p ∧ q.”
Wrong: “~p ∧ ~q” [Not applying De Morgan’s]
Right: “~p ∨ ~q” [Negation flips ∧ to ∨]
Fix: De Morgan’s laws flip both the main operator AND the individual negations.
Mistake 4: Confusing ∨ with AND
Example: “When is p ∨ q true?”
Wrong: “Only when both p and q are true.”
Right: “When at least one of p or q is true.”
Fix: ∨ means OR (at least one). ∧ means AND (both).
Mistake 5: Building Incomplete Truth Tables
Example: “Build a truth table for (p → q) ∧ r.”
Wrong: “Only fills in a few rows.”
Right: “Fills all 2^n rows (8 rows for 3 variables).”
Fix: Truth tables must be exhaustive. Don’t skip rows.
What to Do Next
To master truth tables: Build 20 different truth tables, starting with 2 variables, then 3.
To master implications: Solve 10 problems where you evaluate “If p, then q” with various truth values of p and q.
To master De Morgan’s laws: Practice negating 15 complex statements using these laws.
The Problem Vault Alternative
The Problem Vault has 50+ Mathematical Reasoning problems from actual PUC exams.