MathematicsLow Weightage ★★Class 11
Mathematical Reasoning
Statements, logical connectives (AND, OR, NOT, IMPLIES), truth tables, converse, inverse, contrapositive, and valid arguments — expect 1–2 direct EAPCET questions.
1–2Questions in EAPCET
~1%Paper Weightage
6Core Concepts
2Mistake Traps
Concept Core
Logical statements, connectives, truth tables, and argument validity — the complete logic framework.
Statements and Logical Connectives
A statement (proposition) is a declarative sentence that is either true (T) or false (F), not both.
| Connective | Symbol | Read as | True when |
|---|
| Negation | ¬p (or ~p) | Not p | p is false |
| Conjunction | p ∧ q | p AND q | Both p, q are true |
| Disjunction | p ∨ q | p OR q | At least one is true |
| Implication | p → q | If p then q | Not (p true, q false) |
| Biconditional | p ↔ q | p if and only if q | p, q have same truth value |
Truth Table for Implication p → q
The implication p → q is false only when p is true and q is false. A false premise makes the implication vacuously true.
Related Conditional Statements
For the implication p → q:
Converse: q → p
Inverse: ¬p → ¬q
Contrapositive: ¬q → ¬p
Key equivalences:
• p → q ≡ ¬q → ¬p (contrapositive is logically equivalent to the original)
• p → q ≡ ¬p ∨ q
• Converse and inverse are equivalent to each other, but NOT to the original.
Tautology, Contradiction, and Contingency
Tautology: Always true regardless of truth values of components. Example: p ∨ ¬p (always T).
Contradiction (Fallacy): Always false. Example: p ∧ ¬p (always F).
Contingency: Sometimes true, sometimes false. Most statements.
De Morgan's Laws:
¬(p ∧ q) ≡ ¬p ∨ ¬q
¬(p ∨ q) ≡ ¬p ∧ ¬q
Valid Arguments — Modus Ponens & Tollens
Modus Ponens: If p → q is true and p is true, then q must be true.
Modus Tollens: If p → q is true and q is false, then p must be false.
Hypothetical Syllogism: p → q and q → r → therefore p → r.
An argument is valid when the conclusion logically follows from the premises (regardless of whether premises are actually true).
Quantifiers
Universal quantifier (∀): "For all" — ∀x, P(x) means P is true for every x in the domain.
Existential quantifier (∃): "There exists" — ∃x, P(x) means P is true for at least one x.
Negation of quantifiers:
¬(∀x, P(x)) ≡ ∃x, ¬P(x) [not all → some are not]
¬(∃x, P(x)) ≡ ∀x, ¬P(x) [none → all are not]
Formula Vault
Logic formulas and equivalences for EAPCET.
p → q (Implication)
False ONLY when p=T, q=F
Vacuously true when p is false
Contrapositive
p → q ≡ ¬q → ¬p
Logically equivalent to original
Implication as Disjunction
p → q ≡ ¬p ∨ q
Very useful for truth tables
De Morgan 1
¬(p ∧ q) ≡ ¬p ∨ ¬q
Negation of AND = OR of negations
De Morgan 2
¬(p ∨ q) ≡ ¬p ∧ ¬q
Negation of OR = AND of negations
Tautology
p ∨ ¬p ≡ T (always)
Law of excluded middle
Contradiction
p ∧ ¬p ≡ F (always)
Law of non-contradiction
Biconditional
p ↔ q ≡ (p→q) ∧ (q→p)
True when p and q have same TV
Worked Examples
5 problems — truth tables, contrapositive, De Morgan, tautology, and a classic trap.
EasyWhen is p → q false? Give an example.▾
State the condition under which the implication p → q is false, with an example.
1
p → q is false ONLY when p is TRUE and q is FALSE.
2
Example: 'If it rains, the ground is wet.' (p: it rains; q: ground is wet)
3
False case: it IS raining (p=T) but the ground is NOT wet (q=F) — this contradicts the implication.
4
All other cases (p=F or q=T) make the implication true.
✓ p → q is false only when p is TRUE and q is FALSE
EasyWrite the contrapositive of: 'If n is even, then n² is even'▾
Write the contrapositive of 'If n is even, then n² is even.'
1
Contrapositive of p → q is ¬q → ¬p.
2
p: 'n is even'; q: 'n² is even'
3
¬q: 'n² is not even (n² is odd)'; ¬p: 'n is not even (n is odd)'
4
Contrapositive: 'If n² is odd, then n is odd'
✓ Contrapositive: 'If n² is odd, then n is odd'
MediumNegate: 'All students passed the exam'▾
Write the negation of the statement 'All students passed the exam.'
1
'All students passed' = ∀x, P(x) where P(x) = 'student x passed'
2
Negation: ¬(∀x, P(x)) = ∃x, ¬P(x)
3
In plain English: 'There exists at least one student who did NOT pass the exam'
✓ Negation: 'There exists a student who did not pass the exam'
EAPCET LevelVerify De Morgan's law: ¬(p ∧ q) ≡ ¬p ∨ ¬q using truth table▾
Construct a truth table to verify ¬(p ∧ q) ≡ ¬p ∨ ¬q.
1
Build the truth table for all 4 combinations of T/F for p and q:
2
(T,T): p∧q=T → ¬(p∧q)=F; ¬p∨¬q = F∨F = F ✓
3
(T,F): p∧q=F → ¬(p∧q)=T; ¬p∨¬q = F∨T = T ✓
4
(F,T): p∧q=F → ¬(p∧q)=T; ¬p∨¬q = T∨F = T ✓
5
(F,F): p∧q=F → ¬(p∧q)=T; ¬p∨¬q = T∨T = T ✓
6
Both columns are identical → De Morgan's law verified.
✓ De Morgan's Law verified — both sides have identical truth values in all 4 cases
Trap QuestionThe converse of p→q is logically equivalent to p→q — True or False?▾
Is the converse (q→p) always logically equivalent to the original implication (p→q)?
1
The trap: Students confuse converse with contrapositive.
2
Contrapositive (¬q→¬p): IS logically equivalent to p→q (same truth values in all cases).
3
Converse (q→p): is NOT logically equivalent to p→q in general.
4
Example: 'If it rains, the ground is wet' (p→q). Converse: 'If the ground is wet, it rained' — can be false (someone may have sprinkled water).
5
So converse can be false even when the original is true.
✓ False — converse (q→p) is NOT equivalent to p→q; only the CONTRAPOSITIVE (¬q→¬p) is equivalent
Mistake DNA
2 logic errors from EAPCET distractor analysis.
🔄
Converse ≡ Original Implication
Converse (q→p) and inverse (¬p→¬q) are equivalent to each other, but NOT to the original implication p→q.
❌ Wrong
p→q ≡ q→p (converse) ✗
(can have different
truth values!)
✓ Correct
p→q ≡ ¬q→¬p ✓ (contrapositive)
q→p ≡ ¬p→¬q (converse≡inverse)
Converse ≠ original
Only the contrapositive (¬q→¬p) is logically equivalent to p→q. Converse and inverse are equivalent to each other but form a different equivalence class from the original.
📊
p→q is False When p is False
When p is false, the implication p→q is VACUOUSLY TRUE (regardless of q). This is the hardest logical concept to accept intuitively.
❌ Wrong
p is False, q is True:
p→q is False ✗
(vacuous truth!)
✓ Correct
p→q is False ONLY when:
p=T AND q=F ✓
All other cases: p→q = T ✓
(including F→T and F→F)
Think of it as a promise: 'If it rains, I will carry an umbrella.' If it does NOT rain, I cannot break the promise — so it's vacuously true regardless of whether I carry an umbrella.
Chapter Intelligence
Mathematical reasoning is a quick-win chapter — learn truth tables and contrapositive once, score directly.
EAPCET Weightage (2019–2024)
Implication and truth table~6 Contrapositive vs converse~6 Negation of quantified statements~4 Tautology and contradiction~3 High-Yield PYQ Patterns
When is p→q false?Write contrapositive of given statementIdentify tautology/contradictionNegate a universally quantified statementDe Morgan's law applicationConverse vs contrapositive distinctionIdentify valid argument form
Exam Strategy
- p → q is false ONLY when p=T and q=F. All other combinations give true. This is the #1 tested fact in this chapter.
- Contrapositive (¬q → ¬p) is equivalent to the original. Converse (q → p) is NOT equivalent. Don't confuse them.
- De Morgan's: ¬(p∧q) = ¬p∨¬q; ¬(p∨q) = ¬p∧¬q. These are directly tested — just negate and flip AND↔OR.
- Negating quantifiers: ¬(∀x, P(x)) = ∃x, ¬P(x). "Not all students passed" = "Some student did not pass." Quick and direct.
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