Quantifiers and quantified statements

A quantifier is a word that tells us how many members of a collection must satisfy a stated condition before the resulting sentence counts as true. Two quantifiers are used in mathematics. The symbol $∃$, read 'there exists', is the existential quantifier: it claims that at least one member of the collection meets the condition. The symbol $∀$, read 'for all' or 'for every', is the universal quantifier: it claims that every member of the collection meets the condition. A sentence built from a quantifier applied to a condition on a set is called a quantified statement — it always names a collection and a condition together.

The truth value of a quantified statement follows directly from how strong its claim is:

• A universally quantified statement, $∀x,\ p(x)$, is TRUE only when every element of the collection satisfies the condition. A single element that fails is enough to make the whole statement FALSE.
• An existentially quantified statement, $∃x$ such that $p(x)$, is TRUE the moment even one element satisfies the condition. It is FALSE only when no element in the entire collection satisfies it.

So a universal claim is disproved by a single counterexample, while an existential claim is proved by a single example — the two behave in opposite ways.

Worked example. Let A = {1, 2, 3, 4, 5, 6, 7}. Find the truth value of each statement.

i) $∃x ∈ A$ such that $x − 4 = 3$ — trying $x = 7$ gives $7 − 4 = 3$, so $x = 7 ∈ A$ satisfies the condition. Since one element works, the statement is true (T).

ii) $∀x ∈ A,\ x + 1 > 3$ — this needs $x > 2$ for every element. Checking the smallest element, $x = 1$: $1 + 1 = 2$, which is not greater than 3, so $x = 1$ already fails. One failure is enough to make a universal statement false, so the truth value is F. …