IMAT Mathematics · Sets and Logic

Propositions, Connectives and Quantifiers

A proposition is a statement that is either true or false; connectives (not, and, or, implies) build compound statements, and quantifiers say for all or there exists.

All Sets and Logic notes

Why this matters

No past paper has asked about propositions directly, but the ministry papers write conditions with ∧ and ∀ in their answer options (2024, 2025), and reading an if-then statement correctly is the core of the logical reasoning section.

Concept 1 of 3: Propositions and the connectives not, and, or, implies

A proposition is a sentence with a definite truth value: "7 is prime" is true, "7 is even" is false. A question or a command is not a proposition. Connectives join propositions, and the truth of the result depends only on the truth of the parts.

Definition

Let $p$ and $q$ be propositions.

Or in mathematics is inclusive: $p \vee q$ is true when at least one part is true, including both.
$p \Rightarrow q$ is false in only one case: $p$ true and $q$ false. If $p$ is false, the implication is true whatever $q$ is.
A sentence with an unknown, such as $x > 3$ , becomes a proposition only once $x$ is given (or a quantifier is added).

Connective	Symbol	Read as	True exactly when
Negation	$\neg p$	not p	p is false
Conjunction	$p \wedge q$	p and q	both p and q are true
Disjunction	$p \vee q$	p or q	at least one of p, q is true
Implication	$p \Rightarrow q$	if p then q	it is not the case that p is true and q is false
Biconditional	$p \Leftrightarrow q$	p if and only if q	p and q have the same truth value

Practice this conceptself-check · 3 quick reps

An implication with a false premise is true

p \Rightarrow q

only fails when

p

is true and

q

is false. "If 2 > 3 then the moon is cheese" is a true implication. Options that call it false because the parts are false are wrong.

Concept 2 of 3: Implication, converse, contrapositive and negation

"If it rains, the road is wet" does not mean "if the road is wet, it rained": a hose could have wet it. But it does mean "if the road is dry, it did not rain". The truth table shows this: the contrapositive always agrees with the original, the converse does not.

Definition

From $p \Rightarrow q$ :

Converse $q \Rightarrow p$ and inverse $\neg p \Rightarrow \neg q$ : NOT equivalent to the original.
Contrapositive $\neg q \Rightarrow \neg p$ : always equivalent to the original.
The negation of $p \Rightarrow q$ is $p \wedge \neg q$ ( $p$ happens and $q$ does not), not another implication.
De Morgan: $\neg(p \wedge q)$ is $\neg p \vee \neg q$ , and $\neg(p \vee q)$ is $\neg p \wedge \neg q$ .
In $p \Rightarrow q$ , $p$ is sufficient for $q$ and $q$ is necessary for $p$ . "p only if q" also means $p \Rightarrow q$ .

p	q	p ⇒ q	q ⇒ p (converse)	not q ⇒ not p (contrapositive)
T	T	T	T	T
T	F	F	T	F
F	T	T	F	T
F	F	T	T	T

The third and fifth columns match in every row: an implication and its contrapositive are equivalent. The converse differs.

Practice this conceptself-check · 3 quick reps

The converse is not equivalent

From "if p then q" you may conclude "if not q then not p", but not "if q then p". Options that reverse the implication without negating both parts are the usual wrong answers.

Concept 3 of 3: Quantifiers: for all and there exists

"For all" makes a strong claim, so a single counterexample breaks it. "There exists" makes a weak claim, so a single example proves it. Negating swaps one for the other: the opposite of "everyone passed" is "someone did not pass", not "nobody passed".

Definition

$\forall$ means for all (every); $\exists$ means there exists (at least one); $\exists!$ means there exists exactly one.
$\neg(\forall x,\ P(x))$ is $\exists x : \neg P(x)$ .
$\neg(\exists x : P(x))$ is $\forall x,\ \neg P(x)$ .
A universal statement is disproved by one counterexample; an existential statement is proved by one example.

Statement	In symbols	Its negation
Every x has property P	$\forall x,\ P(x)$	at least one x does not: $\exists x : \neg P(x)$
Some x has property P	$\exists x : P(x)$	no x has it: $\forall x,\ \neg P(x)$
Every real square is at least 0 (true)	$\forall x \in \mathbb{R},\ x^2 \ge 0$	$\exists x \in \mathbb{R} : x^2 < 0$ (false)
Some integer is even and prime (true: 2)	$\exists n \in \mathbb{Z} : n \text{ even} \wedge n \text{ prime}$	every integer is odd or not prime (false)

Practice this conceptself-check · 3 quick reps

The negation of all is not none

"Not every student passed" means at least one failed. "No student passed" is a much stronger claim and is not the negation. IMAT options use this confusion.

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.

Reference tables (3)

Propositions and the connectives not, and, or, implies5 rows

Connective	Symbol	Read as	True exactly when
Negation	$\neg p$	not p	p is false
Conjunction	$p \wedge q$	p and q	both p and q are true
Disjunction	$p \vee q$	p or q	at least one of p, q is true
Implication	$p \Rightarrow q$	if p then q	it is not the case that p is true and q is false
Biconditional	$p \Leftrightarrow q$	p if and only if q	p and q have the same truth value

Implication, converse, contrapositive and negation4 rows

p	q	p ⇒ q	q ⇒ p (converse)	not q ⇒ not p (contrapositive)
T	T	T	T	T
T	F	F	T	F
F	T	T	F	T
F	F	T	T	T

The third and fifth columns match in every row: an implication and its contrapositive are equivalent. The converse differs.

Quantifiers: for all and there exists4 rows

Statement	In symbols	Its negation
Every x has property P	$\forall x,\ P(x)$	at least one x does not: $\exists x : \neg P(x)$
Some x has property P	$\exists x : P(x)$	no x has it: $\forall x,\ \neg P(x)$
Every real square is at least 0 (true)	$\forall x \in \mathbb{R},\ x^2 \ge 0$	$\exists x \in \mathbb{R} : x^2 < 0$ (false)
Some integer is even and prime (true: 2)	$\exists n \in \mathbb{Z} : n \text{ even} \wedge n \text{ prime}$	every integer is odd or not prime (false)

Watch out for (3)

An implication with a false premise is true
→ Propositions and the connectives not, and, or, implies
The converse is not equivalent
→ Implication, converse, contrapositive and negation
The negation of all is not none
→ Quantifiers: for all and there exists

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