MHT-CET Maths · Free notes

Mathematical Logic — MHT-CET Maths

Mathematical Logic is the most self-contained chapter in MHT-CET Maths: 151 PYQs across 2021–2026, and almost all of it is machinery you build from scratch here rather than carry in from other chapters — which makes it the fastest chapter to bank from a cold start. The one exception is real and sits in the very first block: a MHT-CET stem will sometimes make p and q into claims about cube roots of unity, direction cosines or matrix algebra, and then the logic is trivial while the marks turn entirely on settling those claims. That single concept is what makes the opening subtopic harder than it looks (4 of its 20 are HARD); the rest of the chapter genuinely stands alone. 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 23% HARD is badly distributed, and knowing where the difficulty actually sits is worth more than any single formula here: Switching Circuits is 17 q at 53% HARD while Negation is 21 q at 10%, so the smallest subtopic is the expensive one and the block that looks fiddliest is the cheapest. If you are short of time, drill Negation and Finding Truth Values first — 52 questions at 10% and 16% HARD — and leave Switching Circuits until Logical Equivalence and Algebra of Statements is solid, because it depends on it. Work the six subtopics below in order — each genuinely rests on the one before. Every PYQ is tagged — learn the pattern, drill the bank, recover the marks.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.

All 151 past questions MHT-CET Maths strategy MHT-CET home All formulas on one page

Subtopic notes

Formula & revision sheet

17 formulas · 4 reference tables · 38 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 $p \to q \;\text{ agrees with }\; \sim p \vee q \;\text{ in every row} \qquad p \to q = F \;\text{ only when } p = T,\; q = F \qquad F \to q = T$
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 (2)

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.

Reading the Last Column: Equivalence, Tautology, Contradiction, Contingency4 rows

Reading	Last column looks like	Example
Tautology	All T	$p \vee \sim p$
Contradiction	All F	$p \wedge \sim p$
Contingency	Mixed — some T, some F	$p \vee q$ The default case. Most statement patterns are contingencies; the exam asks you to spot the ones that are not.
Logically equivalent $A \equiv B$	Two patterns whose columns match in EVERY row	$p \to q$ and $\sim p \vee q$ ONE disagreeing row destroys equivalence; agreeing in one row proves nothing. The asymmetry is the whole test.

Every 'which of the following is a tautology / is equivalent to' question on this paper is one of these four readings. Subtopic 5 gives you the algebra that reaches them without writing the table out.

Watch out for (10)

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
Reading $\equiv$ as another name for $\leftrightarrow$
→ Reading the Last Column: Equivalence, Tautology, Contradiction, Contingency
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)

Why Equivalence Licenses Substitution · Equivalence and its test $A \equiv B \iff (A \leftrightarrow B) \text{ is a tautology} \iff A \text{ and } B \text{ share every row}$
Classifying a Pattern Without Building Its Table · Classification by last column $\text{tautology} \Rightarrow \text{all } T \qquad \text{contradiction} \Rightarrow \text{all } F \qquad \text{contingency} \Rightarrow \text{mixed}$
Finding the Statement That Makes a Pattern a Tautology · Tautology condition and modus tollens $(X \to r) \text{ is a tautology} \iff X = T \Rightarrow r = T \qquad (p \to q) \wedge \sim q \equiv\; \sim p \wedge \sim q$
The Dual of a Statement Pattern · Dual versus negation $\text{dual}(p \wedge \sim q) = p \vee \sim q \qquad \sim(p \wedge \sim q) \equiv\; \sim p \vee q$

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
→ Why Equivalence Licenses Substitution
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
→ Classifying a Pattern Without Building Its Table
Looking for an r that is true everywhere
→ Finding the Statement That Makes a Pattern a Tautology
Negating the letters when asked for a dual
→ The Dual of a Statement Pattern

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

PYQ weightage by concept

31 concepts · 151 PYQs — where the marks actually sit, so you know what to drill first

Statements, Connectives and Truth Tables12 PYQs · 8%

Concept	PYQs	Share
Settling the Truth Value of a Mathematical Claim	5	3%
Evaluating a Statement Pattern from Given Truth Values	3	2%
The Conditional and Vacuous Truth	2	1%
The Five Logical Connectives	1	1%
Building the Full Truth Table	1	1%
Statements and Truth Valuesfoundation	—	—
Reading the Last Column: Equivalence, Tautology, Contradiction, Contingencyfoundation	—	—

Finding Truth Values of Component Statements15 PYQs · 10%

Concept	PYQs	Share
The Forced Row of a False Conditional	9	6%
Testing the Options Once the Values Are Known	3	2%
Forced Values from a Biconditional	2	1%
Chaining Two Given Truth Values	1	1%

Negation of Statements and Quantifiers14 PYQs · 9%

Concept	PYQs	Share
Negating a Biconditional	5	3%
Negating a Conditional	4	3%
De Morgan Laws for And and Or	2	1%
Negating Quantified Statements	2	1%
Negating a Statement Given in Words	1	1%

Converse, Inverse and Contrapositive17 PYQs · 11%

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

Logical Equivalence and Algebra of Statements16 PYQs · 11%

Concept	PYQs	Share
Simplifying a Statement Pattern	7	5%
Classifying a Pattern Without Building Its Table	4	3%
The Algebra of Statements	2	1%
Finding the Statement That Makes a Pattern a Tautology	2	1%
Why Equivalence Licenses Substitution	1	1%
The Dual of a Statement Patternfoundation	—	—

Switching Circuits10 PYQs · 7%

Concept	PYQs	Share
Writing the Symbolic Form of a Printed Circuit	3	2%
Simplifying a Circuit and Redrawing It	3	2%
Series is And, Parallel is Or	2	1%
Deciding Whether Two Circuits Are Equivalent	2	1%

Test yourself on Mathematical Logic

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Statements, Connectives and Truth Tables

Finding Truth Values of Component Statements

Negation of Statements and Quantifiers

Converse, Inverse and Contrapositive

Logical Equivalence and Algebra of Statements

Switching Circuits

Formula & revision sheet

PYQ weightage by concept

Test yourself on Mathematical Logic

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