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Such That — Definition, Formula & Examples

"Such that" is a phrase used in mathematics to introduce a condition or restriction. It is written using the symbols ∣|, ::, or ∋\ni and means "where the following condition is true."

In set-builder notation and logical statements, "such that" separates the variable(s) being described from the predicate (condition) they must satisfy. For example, {x∈R∣x>0}\{x \in \mathbb{R} \mid x > 0\} denotes the set of all real numbers xx satisfying the condition x>0x > 0.

How It Works

You encounter "such that" whenever a statement needs to specify which elements are being discussed. In set-builder notation, you write a variable on the left side and the condition on the right, separated by ∣| or ::. For instance, {n∈Z:n2<20}\{n \in \mathbb{Z} : n^2 < 20\} reads "the set of all integers nn such that n2n^2 is less than 20." In proofs and logical expressions, the phrase often appears after existential quantifiers: ∃x∈R\exists x \in \mathbb{R} such that x2=9x^2 = 9 means "there exists a real number xx whose square equals 9." The colon :: and vertical bar ∣| are interchangeable in set-builder notation — which one you use is a matter of convention.

Worked Example

Problem: List the elements of the set {x∈Z∣x2<20}\{x \in \mathbb{Z} \mid x^2 < 20\}.
Read the notation: This reads: "the set of all integers x such that x squared is less than 20."
Find the integers: Check which integers have squares less than 20. Since 42=16<204^2 = 16 < 20 but 52=25≥205^2 = 25 \geq 20, the integers that work are −4,−3,−2,−1,0,1,2,3,4-4, -3, -2, -1, 0, 1, 2, 3, 4.
{−4, −3, −2, −1, 0, 1, 2, 3, 4}\{-4,\, -3,\, -2,\, -1,\, 0,\, 1,\, 2,\, 3,\, 4\}
Answer: {−4,−3,−2,−1,0,1,2,3,4}\{-4, -3, -2, -1, 0, 1, 2, 3, 4\}

Why It Matters

You will see "such that" constantly in algebra, precalculus, and any proof-based course like discrete math or linear algebra. Recognizing the symbols ∣| and :: lets you read set definitions, theorems, and formal proofs fluently rather than getting stuck on notation.

Common Mistakes

Mistake: Confusing the vertical bar ∣| meaning "such that" with ∣| meaning "divides" or absolute value.
Correction: Context determines meaning. Inside set-builder braces {…∣…}\{\ldots | \ldots\}, it means "such that." In expressions like 3∣123 \mid 12, it means "divides." Surrounding a number like ∣−5∣|{-5}|, it means absolute value.