MHT-CET Maths · Mathematical Logic

Statements, Connectives and Truth Tables

A statement is a sentence that is definitely true or definitely false; connectives join statements, and a truth table lists the result for every combination of inputs.

All Mathematical Logic notes MHT-CET Maths strategy

Why this matters

This is the machinery every other subtopic in the chapter runs on — build the table, read the last column. 20 PYQs sit here and 4 of them are HARD, which surprises students who expect the opening subtopic to hold none. The difficulty is almost never the logic. It is that a MHT-CET stem will hand you statements like 'the sum of the cube roots of unity is 1' or 'A squared minus B squared equals (A−B)(A+B) for matrices' and expect you to settle their truth from the rest of the syllabus before any connective is touched.

Concept 1 of 7: Statements and Truth Values

Logic only deals with sentences you can definitively call true or false. 'Delhi is the capital of India' qualifies. 'Close the door', 'What time is it?' and 'x + 2 = 5' do not — the first two make no claim, and the third depends on an unknown x.

Definition

A statement (or proposition) is a declarative sentence that is either true or false, but not both. Its truth value is T or F.

Commands, questions, requests and exclamations are not statements.
An open sentence containing a variable ( $x + 2 = 5$ ) is not a statement until the variable is fixed or quantified.
Statements are labelled $p, q, r, s$ ; a simple statement contains no connective, a compound statement is built from simple ones.

Worked example

Which of these are statements? (i) 3 is an even number. (ii) Solve the equation

x^2 = 9

. (iii)

y > 7

. (iv) Every square is a rectangle.

Practise this concept4 quick reps

Treating a false sentence as 'not a statement'

A statement only has to HAVE a truth value, not to be true. '2 + 2 = 5' is a perfectly good statement whose truth value is F. The test is whether the sentence makes a definite claim, never whether the claim is correct.

Concept 2 of 7: The Five Logical Connectives

Five symbols build every compound statement in this chapter. The fastest way to hold them is not the full table but the one row where each is unusual: an AND is true only when both parts are, an OR is false only when both parts are, and a conditional is false only when a true claim leads to a false one.

Definition

Each connective is fully described by when it is false, because each has exactly one interesting case:

$p \wedge q$ (and) is true only when both are true.
$p \vee q$ (or) is inclusive — true when at least one holds, including both.
$p \to q$ (if–then) is false only for $T \to F$ .
$p \leftrightarrow q$ (if and only if) is true exactly when both sides match.
$\sim p$ (not) simply flips the value.

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.

Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2023 · 4th May Shift 2 · Q126Easy

Example 2 · Mathematical Logic · Statements, Connectives and Truth Tables

If the statements p, q and r have the truth values F, T, F respectively, then the truth values of the statement patterns

(p\wedge\sim q)\rightarrow r

and

(p\vee q)\rightarrow r

are respectively

Reading 'or' as exclusive

In logic

p \vee q

is TRUE when p and q are both true. A student who reads 'or' as 'one or the other but not both' will mark the both-true row F and get every disjunction question wrong by exactly one row.

Concept 3 of 7: The Conditional and Vacuous Truth

A conditional is a promise: 'if p, then q'. You only break the promise by having p happen and q fail. If p never happens, you have not broken anything — so the promise counts as kept. That is why 'If 3 + 2 = 7, then the earth is flat' is a TRUE statement.

Definition

$p \to q$ is false in exactly one row: $p$ true and $q$ false. In the other three rows it is true.

When $p$ is false, $p \to q$ is true no matter what $q$ says. This is called vacuous truth.
When $q$ is true, $p \to q$ is true no matter what $p$ says.
$p \to q$ has the same truth value as $\sim p \vee q$ in every row, which is the form you will use to simplify. The next-but-one concept gives that relationship its name and its symbol.

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

$p$ the antecedent (hypothesis)
$q$ the consequent (conclusion)
$F \to q$ vacuously true — a false antecedent makes the whole conditional true

Worked example

Decide the truth value of each: (A) If

2 + 2 = 5

, then Mumbai is in India. (B) If

2 + 2 = 4

, then Mumbai is in Pakistan. (C) If

2+2=5

, then

3+3=9

Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 25 April Shift II · Q105Moderate

Example 3 · Mathematical Logic · Statements, Connectives and Truth Tables

Consider the following three statements (A) If

3 + 2 = 7

then

4 + 3 = 8

. (B) If

5 + 2 = 7

then earth is flat. (C) If both (A) and (B) are true then

5 + 6 = 11

. Which of the following statements is correct?

Calling a conditional false because its parts are false

'If 3 + 2 = 7, then the earth is flat' has a false antecedent AND a false consequent, and the whole statement is true. Students reject it because both halves are nonsense. Only the

T \to F

pattern makes a conditional false — check the PATTERN, not the plausibility.

Chained stems that hide a false antecedent

A recurring MHT-CET stem gives statements (A) and (B), then (C) = 'If both (A) and (B) are true, then …'. If either (A) or (B) is false, (C)'s antecedent is false and (C) is automatically true — however absurd its consequent. Evaluate (A) and (B) first; the answer to (C) usually falls out with no work at all.

Drill 1 more on the conditional and vacuous truth

Concept 4 of 7: Evaluating a Statement Pattern from Given Truth Values

When the question hands you the truth values of p, q and r, you are not building a table at all — you are substituting numbers into a formula. Work strictly from the innermost bracket outwards and write the value above each piece as you go.

Definition

To evaluate a compound statement for one fixed assignment of truth values:

Replace every letter by its given value.
Resolve every $\sim$ first, then work outwards from the innermost bracket.
Apply one connective at a time, writing the intermediate value down rather than holding it in your head.

A single assignment gives one row, not a table — no other row can change the answer.

Worked example

p

is true,

q

is false and

r

is true, find the truth value of

[\,(p \wedge \sim q) \vee r\,] \to (q \wedge r)

Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2024 · 9th May Shift 1 · Q148Hard

Example 4 · Mathematical Logic · Statements, Connectives and Truth Tables

If truth values of statements

p, q

are true, and

r, s

are false, then the truth values of the following statement patterns are respectively. a:

\sim(p\wedge\sim r)\vee(\sim q\vee s)

(\sim q\wedge\sim r)\leftrightarrow(p\vee s)

(\sim p\vee q)\to(r\wedge\sim s)

Resolving the negation last instead of first

\sim(p \wedge q)

and

\sim p \wedge q

are different statements. The tilde binds only as far as its bracket reaches, so decide what the

\sim

is sitting on BEFORE you substitute anything.

Drill 2 more on evaluating a statement pattern from given truth values

Concept 5 of 7: Building the Full Truth Table

When no truth values are given, you must check every possibility. Two letters need 4 rows, three letters need 8 — and the standard row order (TT, TF, FT, FF) matters, because MHT-CET options are given as a last column like 'TFTF' and you have to match them position by position.

Definition

A statement pattern in $n$ distinct letters needs $2^{n}$ rows.

Fill the input columns in the standard order: for two letters, $TT,\,TF,\,FT,\,FF$ .
Add one column per intermediate piece, working from the innermost bracket outwards.
The last column is the statement's truth value in each row, and it is what the options quote.

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}

$n$ the number of DISTINCT statement letters, not the number of connectives

Worked example

Construct the truth table for

(p \vee q) \to p

and write its last column.

Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 19 April Shift II · Q101Moderate

Example 5 · Mathematical Logic · Statements, Connectives and Truth Tables

The last column in the truth table of the statement pattern

\lbrack p\rightarrow (q\land \sim p)\rbrack \vee \lbrack(p\vee \sim q) \land p\rbrack

Counting connectives instead of letters

The row count is

2^{n}

where

n

is the number of DISTINCT letters.

[p \to (q \wedge \sim p)] \vee [(p \vee \sim q) \wedge p]

has six connectives and only two letters, so it needs 4 rows, not 64.

Matching the last column in the wrong row order

Options are quoted as a bare string like 'TTFT'. That only matches your table if you filled the input columns in the standard

TT,\,TF,\,FT,\,FF

order. Writing the rows in a different order gives a correct table and the wrong option.

Concept 6 of 7: Reading the Last Column: Equivalence, Tautology, Contradiction, Contingency

You now know how to build the table. Everything the rest of this chapter asks you to do is a question about its LAST COLUMN — is it all T, all F, mixed, or the same as some other pattern's last column? Those four readings have names, and they are used constantly from here on.

Definition

Four readings of one column, and one symbol:

A tautology is true in every row.
A contradiction is false in every row.
A contingency is neither — at least one T and at least one F.
Two patterns are logically equivalent, written $A \equiv B$ , when their last columns match row for row.

The symbol $\equiv$ is not a connective and never appears inside a statement: $\leftrightarrow$ is evaluated row by row and produces a column, while $\equiv$ is a claim ABOUT two finished columns. The two are linked — $A \equiv B$ exactly when $A \leftrightarrow B$ is a tautology.

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.

Practise this conceptself-check · 4 quick reps

Reading $\equiv$ as another name for $\leftrightarrow$

They are different kinds of object.

p \leftrightarrow q

is a statement pattern with its own truth table, true in some rows and false in others.

A \equiv B

is a verdict on two whole tables, and it is either right or wrong — it has no rows. The bridge between them is that

A \equiv B

holds exactly when the pattern

A \leftrightarrow B

is a tautology.

Concept 7 of 7: Settling the Truth Value of a Mathematical Claim

Some MHT-CET stems make p, q and r be mathematical assertions from other chapters — direction cosines, cube roots of unity, binomial sums, matrix algebra. The logic in these questions is trivial. The entire difficulty is deciding whether each claim is true before you touch a connective.

Definition

For stems of the form 'p: <some mathematical claim>, q: <another> … which of the following is correct?':

Settle each letter's truth value independently and first, using the relevant chapter's own facts.
Write T or F beside each letter before reading the options.
Only then substitute into the options, which is ordinary one-row evaluation.

A single misjudged claim flips the whole answer, so this is where the marks are actually lost.

Worked example

Let p: the sum of the cube roots of unity is 1. Let q:

\sqrt{2}

is irrational. Find the truth value of

(\sim p) \wedge q

Practise this conceptself-check · 4 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 19 April Shift I · Q148Hard

Example 7 · Mathematical Logic · Statements, Connectives and Truth Tables

Consider the three statements

p:\forall n \in\mathbb{N},10n - 3

is a prime number, when n is not divisible by 3 .

q:\frac{2}{\sqrt{3}},\frac{- 2}{\sqrt{3}},\frac{- 1}{\sqrt{3}}

are the direction cosines of a directed line.

r:\sin x

is an increasing function in the interval

\left\lbrack \frac{-\pi}{2},\frac{\pi}{2} \right\rbrack

. Then which of the following statement pattern has truth value true?

Doing the logic correctly on a misjudged claim

These questions are graded on the mathematics, not the logic. If you record 'there are 26 primes below 100' as true, every connective afterwards is applied flawlessly to the wrong input and the answer is wrong. Settle each letter before reading the options.

Assuming a 'for all' claim is true because it works for small n

A universally quantified claim fails on ONE counterexample. '

10n - 3

is prime whenever

n

is not divisible by 3' survives

n = 1, 2, 4

and dies at

n = 8

, where

77 = 7 \times 11

. Hunt for the counterexample rather than confirming the pattern.

Drill 4 more on settling the truth value of a mathematical claim

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

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.

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

Test yourself on Mathematical Logic

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