MHT-CET Maths · Teaching notes

Mathematical Logic — MHT-CET Maths

Mathematical Logic is the most self-contained chapter in MHT-CET Maths: 88 PYQs across 2021–2025 that borrow almost nothing from the rest of the syllabus, which makes it the fastest chapter to bank from a cold start. It is also the one chapter with its own execution mode — roughly 70% of its stems hand you four claims to adjudicate rather than a problem to solve. Its 31% HARD is badly distributed, and knowing where the difficulty actually sits is worth more than any single formula here: Switching Circuits is 12 q at 67% HARD while Negation is 14 q at 14%, so the smallest subtopic is the expensive one and the block that looks fiddliest is the cheapest. The chapter teaches in six movements, each resting on the one before: (1) Statements, Connectives and Truth Tables — what counts as a statement, the five connectives, vacuous truth, and building the table; (2) Finding Truth Values of Component Statements — the signature MHT-CET move run backwards, where you are told the pattern is false and asked for p, q and r; (3) Negation of Statements and Quantifiers — De Morgan, negating a conditional and a biconditional, and flipping quantifiers; (4) Converse, Inverse and Contrapositive — the three relatives of a conditional, only one of which is equivalent to it; (5) Logical Equivalence and Algebra of Statements — the simplification laws, duals, and classifying a pattern as tautology, contradiction or contingency; (6) Switching Circuits — series is AND, parallel is OR, and everything you already know applies. Every PYQ is tagged — learn the pattern, drill the bank, recover the marks.

Every subtopic, worked example, formula and trap in one printable document — answers shown, ready to share.

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Subtopic notes

PYQ weightage by concept

30 concepts · 88 PYQs — where the marks actually sit, so you know what to drill first

Statements, Connectives and Truth Tables13 PYQs · 15%

Concept	PYQs	Share
Settling the Truth Value of a Mathematical Claim	5	6%
Evaluating a Statement Pattern from Given Truth Values	3	3%
The Conditional and Vacuous Truth	2	2%
Building the Full Truth Table	2	2%
The Five Logical Connectives	1	1%
Statements and Truth Valuesfoundation	—	—

Finding Truth Values of Component Statements16 PYQs · 18%

Concept	PYQs	Share
The Forced Row of a False Conditional	9	10%
Testing the Options Once the Values Are Known	3	3%
Forced Values from a Biconditional	2	2%
Chaining Two Given Truth Values	2	2%

Negation of Statements and Quantifiers14 PYQs · 16%

Concept	PYQs	Share
Negating a Biconditional	5	6%
Negating a Conditional	4	5%
De Morgan Laws for And and Or	2	2%
Negating Quantified Statements	2	2%
Negating a Statement Given in Words	1	1%

Converse, Inverse and Contrapositive17 PYQs · 19%

Concept	PYQs	Share
The Three Relatives of a Conditional	7	8%
Stacked Operations: Negation of a Contrapositive, Contrapositive of an Inverse	4	5%
Only the Contrapositive Shares the Truth Value	3	3%
Converting to Conditional Form First	2	2%
Necessary and Sufficient Condition Language	1	1%

Logical Equivalence and Algebra of Statements16 PYQs · 18%

Concept	PYQs	Share
Simplifying a Statement Pattern	7	8%
Tautology, Contradiction and Contingency	4	5%
The Algebra of Statements	2	2%
Finding the Statement That Makes a Pattern a Tautology	2	2%
What Logical Equivalence Means	1	1%
The Dual of a Statement Patternfoundation	—	—

Switching Circuits12 PYQs · 14%

Concept	PYQs	Share
Writing the Symbolic Form of a Printed Circuit	4	5%
Simplifying a Circuit and Redrawing It	3	3%
Deciding Whether Two Circuits Are Equivalent	3	3%
Series is And, Parallel is Or	2	2%

Formula & revision sheet

17 formulas · 3 reference tables · 37 gotchas across all subtopics — the exam-eve cheat-sheet

Statements, Connectives and Truth Tables

Formulas (2)

The Conditional and Vacuous Truth · The conditional and its disjunction form $p \to q \equiv\; \sim p \vee q \qquad p \to q = F \iff (p = T \text{ and } q = F) \qquad F \to q \equiv T$
Building the Full Truth Table · Number of rows in a truth table $\text{rows} = 2^{n} \qquad n = 2 \Rightarrow 4 \text{ rows}, \qquad n = 3 \Rightarrow 8 \text{ rows}$

Reference tables (1)

The Five Logical Connectives5 rows

Connective	Symbol	Read as	Value
Negation	$\sim p$	not p	Flips: $\sim T = F$ , $\sim F = T$
Conjunction	$p \wedge q$	p and q	T only when both p and q are T
Disjunction	$p \vee q$	p or q	F only when both p and q are F Inclusive OR: 'p or q' is TRUE when both hold. Everyday English often means the exclusive one; logic never does.
Conditional	$p \to q$	if p then q	F only when p is T and q is F The single most-tested row in the chapter. A conditional with a FALSE antecedent is TRUE, whatever the consequent says.
Biconditional	$p \leftrightarrow q$	p if and only if q	T when p and q have the SAME value

Every question in this chapter is this table applied repeatedly. Learn the Value column and the rest is bookkeeping.

Watch out for (9)

Treating a false sentence as 'not a statement'
→ Statements and Truth Values
Reading 'or' as exclusive
→ The Five Logical Connectives
Calling a conditional false because its parts are false
→ The Conditional and Vacuous Truth
Chained stems that hide a false antecedent
→ The Conditional and Vacuous Truth
Resolving the negation last instead of first
→ Evaluating a Statement Pattern from Given Truth Values
Counting connectives instead of letters
→ Building the Full Truth Table
Matching the last column in the wrong row order
→ Building the Full Truth Table
Doing the logic correctly on a misjudged claim
→ Settling the Truth Value of a Mathematical Claim
Assuming a 'for all' claim is true because it works for small n
→ Settling the Truth Value of a Mathematical Claim

Finding Truth Values of Component Statements

Formulas (2)

The Forced Row of a False Conditional · The one false row of a conditional $X \to Y = F \iff X = T \;\text{ and }\; Y = F \qquad (A \wedge B) = T \Rightarrow A = B = T \qquad (A \vee B) = F \Rightarrow A = B = F$
Forced Values from a Biconditional · The matching rule $X \leftrightarrow Y = T \iff X = Y \qquad X \leftrightarrow Y = F \iff X \neq Y$

Watch out for (5)

Trying to work backwards from a TRUE conditional
→ The Forced Row of a False Conditional
Hunting for p when p is not determined
→ Forced Values from a Biconditional
Starting with the given that forces least
→ Chaining Two Given Truth Values
Evaluating a consequent you never needed
→ Testing the Options Once the Values Are Known
Reporting the truth values when the question asked for an option
→ Testing the Options Once the Values Are Known

Negation of Statements and Quantifiers

Formulas (4)

De Morgan Laws for And and Or · De Morgan laws $\sim(p \wedge q) \equiv\; \sim p \vee \sim q \qquad \sim(p \vee q) \equiv\; \sim p \wedge \sim q \qquad \sim(\sim p) \equiv p$
Negating a Conditional · Negation of an implication $\sim(p \to q) \equiv p \wedge \sim q$
Negating a Biconditional · Negation of a biconditional $\sim(p \leftrightarrow q) \equiv (p \wedge \sim q) \vee (\sim p \wedge q) \qquad \sim(p \leftrightarrow \sim q) \equiv p \leftrightarrow q$
Negating Quantified Statements · Quantifier negation $\sim(\forall x,\, p(x)) \equiv \exists x,\; \sim p(x) \qquad \sim(\exists x,\, p(x)) \equiv \forall x,\; \sim p(x) \qquad \sim(x \geq M) \equiv (x < M)$

Watch out for (6)

Negating both parts but keeping the connective
→ De Morgan Laws for And and Or
Negating an implication as another implication
→ Negating a Conditional
Negating both sides of a biconditional
→ Negating a Biconditional
Assigning a negative statement to a letter
→ Negating a Statement Given in Words
Flipping the quantifier but leaving the predicate alone
→ Negating Quantified Statements
Forgetting that the inequality changes
→ Negating Quantified Statements

Converse, Inverse and Contrapositive

Formulas (4)

Only the Contrapositive Shares the Truth Value · The equivalence pairs $p \to q \equiv\; \sim q \to \sim p \qquad q \to p \equiv\; \sim p \to \sim q \qquad p \to q \not\equiv q \to p$
Converting to Conditional Form First · Conditional law, used in reverse $\sim p \vee q \equiv p \to q \qquad p \vee q \equiv\; \sim p \to q \qquad p \to q \equiv\; \sim q \to \sim p$
Stacked Operations: Negation of a Contrapositive, Contrapositive of an Inverse · The chain that collapses $\text{contrapositive}(\text{inverse}(p \to q)) \equiv q \to p \qquad \sim(\text{contrapositive}(p \to q)) \equiv\; \sim q \wedge p$
Necessary and Sufficient Condition Language · Equivalent phrasings of one conditional $p \to q \;\equiv\; (p \text{ only if } q) \;\equiv\; (q \text{ necessary for } p) \;\equiv\; (p \text{ sufficient for } q) \;\equiv\; \sim q \to \sim p$

Reference tables (1)

The Three Relatives of a Conditional5 rows

Form	Symbolic	Built by	Equivalent to original?
Original	$p \to q$	—	Yes, trivially
Converse	$q \to p$	Swap the two parts	No
Inverse	$\sim p \to \sim q$	Negate both parts, keep the order	No
Contrapositive	$\sim q \to \sim p$	Swap and negate both	Yes — always The only equivalent relative, and the one the paper asks about most. A statement and its contrapositive always share a truth value.
Converse and inverse	$q \to p$ and $\sim p \to \sim q$	Each is the contrapositive of the other	Equivalent to EACH OTHER, not to the original

Memorise the last column. Most option lists contain all three relatives, so knowing the forms is not enough — you must know which one is being asked for.

Watch out for (6)

Offering the converse where the contrapositive was asked
→ The Three Relatives of a Conditional
Assuming the converse follows from the original
→ Only the Contrapositive Shares the Truth Value
Negating the wrong part when converting an OR
→ Converting to Conditional Form First
Applying the second operation to the original statement
→ Stacked Operations: Negation of a Contrapositive, Contrapositive of an Inverse
Reading 'p only if q' as 'if q then p'
→ Necessary and Sufficient Condition Language
Swapping necessary and sufficient
→ Necessary and Sufficient Condition Language

Logical Equivalence and Algebra of Statements

Formulas (4)

What Logical Equivalence Means · Equivalence and its test $A \equiv B \iff (A \leftrightarrow B) \text{ is a tautology} \iff A \text{ and } B \text{ share every row}$
Tautology, Contradiction and Contingency · Classification by last column $\text{tautology} \Rightarrow \text{all } T \qquad \text{contradiction} \Rightarrow \text{all } F \qquad \text{contingency} \Rightarrow \text{mixed}$
The Dual of a Statement Pattern · Dual versus negation $\text{dual}(p \wedge \sim q) = p \vee \sim q \qquad \sim(p \wedge \sim q) \equiv\; \sim p \vee q$
Finding the Statement That Makes a Pattern a Tautology · Tautology condition and modus tollens $(X \to r) \text{ is a tautology} \iff X = T \Rightarrow r = T \qquad (p \to q) \wedge \sim q \equiv\; \sim p \wedge \sim q$

Reference tables (1)

The Algebra of Statements10 rows

Law	With AND	With OR
Commutative	$p \wedge q \equiv q \wedge p$	$p \vee q \equiv q \vee p$
Associative	$(p \wedge q) \wedge r \equiv p \wedge (q \wedge r)$	$(p \vee q) \vee r \equiv p \vee (q \vee r)$
Distributive	$p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$	$p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$ Both directions are legal here, unlike ordinary arithmetic where only one distribution holds.
Identity	$p \wedge T \equiv p$	$p \vee F \equiv p$
Domination	$p \wedge F \equiv F$	$p \vee T \equiv T$
Complement	$p \wedge \sim p \equiv F$	$p \vee \sim p \equiv T$ The engine of most simplifications: spot a letter meeting its own negation and a whole branch collapses to F or T.
Idempotent	$p \wedge p \equiv p$	$p \vee p \equiv p$
Absorption	$p \wedge (p \vee q) \equiv p$	$p \vee (p \wedge q) \equiv p$ The whole bracket vanishes. Worth memorising by shape: a letter outside meeting itself inside swallows the rest.
De Morgan	$\sim(p \wedge q) \equiv \;\sim p \vee \sim q$	$\sim(p \vee q) \equiv \;\sim p \wedge \sim q$
Conditional	$\sim(p \to q) \equiv p \wedge \sim q$	$p \to q \equiv \;\sim p \vee q$ Always apply this first. The other laws cannot see through an arrow.

Ten laws cover every simplification the paper sets. Complement, distributive and absorption account for most of the work.

Watch out for (6)

Checking one row and declaring equivalence
→ What Logical Equivalence Means
Distributing only one way
→ The Algebra of Statements
Simplifying around an arrow instead of clearing it
→ Simplifying a Statement Pattern
Calling a contingency a tautology after checking two rows
→ Tautology, Contradiction and Contingency
Negating the letters when asked for a dual
→ The Dual of a Statement Pattern
Looking for an r that is true everywhere
→ Finding the Statement That Makes a Pattern a Tautology

Switching Circuits

Formulas (1)

Series is And, Parallel is Or · The translation rule $\text{series} \;\longrightarrow\; p \wedge q \qquad \text{parallel} \;\longrightarrow\; p \vee q \qquad S_1' \;\longrightarrow\; \sim p$

Watch out for (5)

Reading a branch before finishing it
→ Series is And, Parallel is Or
Giving a primed switch its own letter
→ Writing the Symbolic Form of a Printed Circuit
Trying to simplify by staring at the picture
→ Simplifying a Circuit and Redrawing It
Deleting a repeated switch instead of factoring it
→ Simplifying a Circuit and Redrawing It
Matching circuits by how they look
→ Deciding Whether Two Circuits Are Equivalent