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. 13 PYQs sit here and 38% of them are HARD, which is higher than the chapter average and surprises students who expect the opening subtopic to be the easiest. 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 6

Statements and Truth Values

Intuition

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.

Practice 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 6

The Five Logical Connectives

Intuition

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.

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 2Mathematical LogicEASY

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

[Q126 · 4th May Shift 2 · 2023]

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 6

The Conditional and Vacuous Truth

Intuition

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.
Equivalently $p \to q \equiv \;\sim p \vee q$ , which is the form you use to simplify.

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

pthe antecedent (hypothesis)
qthe 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

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 3Mathematical LogicMODERATE

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?

[Q105 · 25 April Shift II · 2025]

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 6

Evaluating a Statement Pattern from Given Truth Values

Intuition

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)

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 4Mathematical LogicHARD

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)

[Q148 · 9th May Shift 1 · 2024]

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 6

Building the Full Truth Table

Intuition

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}

nthe 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.

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 5Mathematical LogicHARD

The last column in the truth table of the statement pattern

[p \to (q \land \sim p)] \lor [(p \lor \sim q) \land p]

[Shift || · 2025]

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.

Drill 1 more on building the full truth table

Concept 6 of 6

Settling the Truth Value of a Mathematical Claim

Intuition

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

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 6Mathematical LogicHARD

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?

[Q148 · 19 April Shift I · 2025]

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
$p \to q \equiv\; \sim p \vee q \qquad p \to q = F \iff (p = T \text{ and } q = F) \qquad F \to q \equiv T$
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

Drill every past-year question on this subtopic

13 questions from the bank — paginated, with cart and Word-export support.

Drill the 13 questions

Drill every past-year question on this subtopic

Related notes