In set theory, a set (or class) A is transitive, if

• whenever x ∈ A, and y ∈ x, then y ∈ A, or, equivalently,
• whenever x ∈ A, and x is not an urelement, then x is a subset of A.

The transitive closure of a set A is the smallest (with respect to inclusion) transitive set B which contains A. Suppose one is given a set X, then the transitive closure of X is:

$\backslash cup\; \backslash \{\; X,\; \backslash cup\; X,\; \backslash cup\; \backslash cup\; X,\; \backslash cup\; \backslash cup\; \backslash cup\; X,\; \backslash cup\; \backslash cup\; \backslash cup\; \backslash cup\; X,\; ...\; \backslash \}$.

Transitive classes are often used for construction of interpretations of set theory in itself, usually called inner models. The reason is that properties defined by bounded formulas are absolute for transitive classes.

An ordinal number may be defined as a transitive set whose members are also transitive.

See also:

• Union (set theory)

Set theory

Ensemble transitif