← Mathematics — Year 1/Discrete Mathematics: Logic, Sets and Proof

Predicates and Quantifiers

≈ 20 min

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A predicate is a statement whose truth depends on a variable, like $P(x):\ x^2 \ge 0$ . Quantifiers turn predicates into definite claims:

Universal $\forall x\,P(x)$ : 'for all $x$ , $P(x)$ holds.'
Existential $\exists x\,P(x)$ : 'there exists an $x$ with $P(x)$ .'

Negation swaps the quantifier and negates the inside:

\neg\big(\forall x\,P(x)\big) \equiv \exists x\,\neg P(x), \qquad \neg\big(\exists x\,P(x)\big) \equiv \forall x\,\neg P(x).

So the opposite of 'all learners passed' is not 'no learner passed' — it is 'at least one learner did not pass.' A single counterexample is enough to defeat a universal claim.

Careful

The negation of 'all $A$ are $B$ ' is 'some $A$ is not $B$ ', never 'no $A$ is $B$ '. Confusing some-not with none is one of the most common reasoning errors.

Which statement is true?

$\forall x \in \mathbb{Z},\ x > 0$

$\exists x \in \mathbb{Z},\ x^2 = 4$

$\forall x \in \mathbb{Z},\ x^2 = 4$

$\exists x \in \mathbb{Z},\ x^2 = -1$

Is $\forall x \in \mathbb{R},\ x^2 \ge 0$ true or false?

True — a real number squared is never negative

False — negative $x$ gives negative $x^2$

False — $x=0$ breaks it

True only for $x \ge 0$

The negation of $\forall x\, P(x)$ is:

$\forall x\, \neg P(x)$

$\exists x\, \neg P(x)$

$\neg\exists x\, P(x)$

$\forall x\, P(x)$

A principal claims: 'Every learner in the school passed.' In one short sentence, state what must be true for this claim to be false.

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