---
title: "Vertical bar mathematical notation"
description: "A context-sensitive delimiter used for conditioning, magnitude, size, determinants, evaluation, divisibility, and set-builder clauses."
canonical_url: "https://fanout.sh/labs/math-decoder/symbol/vertical-bar"
md_url: "https://fanout.sh/labs/math-decoder/symbol/vertical-bar.md"
last_updated: "2026-08-09"
access: "public"
---

# Vertical bar mathematical notation

A context-sensitive delimiter used for conditioning, magnitude, size, determinants, evaluation, divisibility, and set-builder clauses.

## Public overview

Conditioning: Restricts a probability or expectation to cases where the expression on the right is known to hold. Example: Measure how likely A is among cases where B is true.

Absolute value: Measures a scalar's distance from zero, ignoring sign. Example: Negative three is three units away from zero.

Set cardinality: Counts the elements in a finite set. Example: The set contains three elements.

Determinant: Returns the signed volume-scaling factor of a square linear transformation. Example: The matrix's determinant is the product of one diagonal minus the other.

Evaluation at a boundary: Substitutes the value written beside the bar into the preceding expression. Example: Substitute three for x in x squared.

Divisibility: States that the number on the left divides the number on the right exactly. Example: Twelve is an integer multiple of three.

Such that: Separates an element description from the condition it must satisfy in set-builder notation. Example: Collect every positive real number.

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