Set-builder Notation - Building Sets

Building Sets

Let Φ(x) be a formula in which x appears free. Set builder notation has the form {x : Φ(x)} (or {x | Φ(x)} according to the international standard ISO 31-11 using the vertical bar instead of the colon), denoting the set of all individuals in the universe of discourse satisfying the formula Φ(x), that is, the set whose members are every individual y such that Φ(y) is true: formally, the extension of the predicate. Sometimes the universe of discourse is established within the notation; writing {x ∈ U : Φ(x)} establishes that the universe of discourse is U, for purposes of the set being built. Set builder notation binds the variable x and must be used with the same care applied to variables bound by quantifiers.

Examples (the universe of discourse can be taken to be the set of real numbers, where not specified inside the notation):

• is the set ,
• is the set of all positive real numbers.
• is the set of all even natural numbers,
• is the set of rational numbers; that is, numbers that can be written as the ratio of two integers.
• Thus, e.g., etc. (n.b.: in the case of sets, the order is not important; could be used). As an example,

The sign stands for and, requiring both conditions be fulfilled simultaneously. It is often replaced by a comma (,) semicolon (;) or written out as and. The sign denotes set membership, and can be read as "in". The sign stands for "there exists" and is formally known as Existential quantification.