yields 1 if expr is True and 0 if it is False.


  • Boole[expr] remains unchanged if expr is neither True nor False.
  • Boole[expr] is effectively equivalent to If[expr,1,0].
  • Boole is Listable.
  • Integrate[f Boole[pred],] can be used to integrate f over the region in which pred is True.


open allclose all

Basic Examples  (3)

Convert a list of truth values to integers:

Find the area of the disc defined by the inequality :

Applications  (4)

Find the area of a region defined by an inequality:

Plot the region:

Find the volume of a solid defined by an inequality:

Plot the solid:

Find the area of the intersection of a circle with a parametric radius and a square:

Convert a table of True,False values to 1,0 for plotting:

Properties & Relations  (7)

The inverse of Boole:

Use PiecewiseExpand to convert Boole to Piecewise:

Correspondence between logic operations on conditions and arithmetic operations on Boole:

Use Refine, Simplify, and FullSimplify to transform expressions involving Boole:

Reduce equations and optimize functions involving Boole:

Use Boole as a characteristic function for integration problems:

Boole can be represented as a DifferenceRoot in certain cases:

Introduced in 2004
Updated in 2007