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Universal Sentence — Definition, Formula & Examples

A universal sentence is a logical statement that claims something is true for every element in a given domain. It uses the universal quantifier ∀\forall (read "for all") to express that a property or condition applies without exception.

A universal sentence is a closed formula of the form ∀x P(x)\forall x\, P(x), where P(x)P(x) is a predicate and the quantifier ∀\forall binds the variable xx over a specified domain of discourse DD. The sentence is true if and only if P(a)P(a) holds for every element a∈Da \in D.

Key Formula

∀x∈D,  P(x)\forall x \in D,\; P(x)
Where:
  • ∀\forall = Universal quantifier, read as "for all" or "for every"
  • xx = A variable ranging over the domain
  • DD = The domain of discourse (the set of objects under consideration)
  • P(x)P(x) = A predicate — a statement whose truth depends on x

How It Works

To evaluate a universal sentence, you check whether the predicate is satisfied by every object in the domain. If even one object fails to satisfy it, the sentence is false — that single failure is a counterexample. In practice, universal sentences often take the form of conditionals: ∀x (Q(x)→R(x))\forall x\,(Q(x) \rightarrow R(x)), meaning "for all xx, if Q(x)Q(x) then R(x)R(x)." Proving a universal sentence typically requires a general argument (such as a direct proof or proof by contradiction) that covers all cases, whereas disproving it requires only one counterexample.

Example

Problem: Let the domain be all integers. Determine the truth value of the universal sentence: "For every integer nn, n2≥0n^2 \geq 0."
Formalize: Write the sentence in symbolic form with the universal quantifier.
∀n∈Z,  n2≥0\forall n \in \mathbb{Z},\; n^2 \geq 0
Check cases: Consider representative values: if n=3n = 3, then n2=9≥0n^2 = 9 \geq 0. If n=0n = 0, then n2=0≥0n^2 = 0 \geq 0. If n=−5n = -5, then n2=25≥0n^2 = 25 \geq 0. Squaring any integer always yields a non-negative result.
Conclude: Because no integer can produce a negative square, the predicate holds for every element of the domain.
Answer: The universal sentence ∀n∈Z,  n2≥0\forall n \in \mathbb{Z},\; n^2 \geq 0 is true.

Why It Matters

Universal sentences are the backbone of mathematical theorems — nearly every theorem asserts that something holds for all objects meeting certain conditions. Learning to read, write, and negate them correctly is essential in proof-based courses like discrete mathematics, real analysis, and abstract algebra.

Common Mistakes

Mistake: Believing that checking a few examples proves a universal sentence is true.
Correction: Examples can only disprove a universal sentence (via counterexample). To prove one, you need a general argument that covers every element in the domain.