The Mathematica Book Online: Advanced Mathematics in Mathematica

The Mathematica Book Online

Advanced Mathematics in Mathematica

Algebraic Manipulation

3.3.9 Advanced Topic: Logical and Piecewise Functions

Nested logical and piecewise functions can be expanded out much like nested arithmetic functions. You can do this using LogicalExpand and PiecewiseExpand.

LogicalExpand[expr] expand out logical functions in expr
PiecewiseExpand[expr] expand out piecewise functions in expr
PiecewiseExpand[expr, assum] expand out with the specified assumptions

Expanding out logical and piecewise functions.

LogicalExpand puts logical expressions into a standard disjunctive normal form (DNF), consisting of an OR of ANDs.

By default, Mathematica leaves this expression unchanged.

In[1]:= (a || b) && c

Out[1]=

LogicalExpand expands this into an OR of ANDs.

In[2]:= LogicalExpand[%]

Out[2]=

LogicalExpand works on all logical functions, always converting them into a standard OR of ANDs form. Sometimes the results are inevitably quite large.

Xor can be expressed as an OR of ANDs.

In[3]:= LogicalExpand[Xor[a, b, c]]

Out[3]=

Any collection of nested conditionals can always in effect be flattened into a piecewise normal form consisting of a single Piecewise object. You can do this in Mathematica using PiecewiseExpand.

By default, Mathematica leaves this expression unchanged.

In[4]:= If[x > 0, If[x < 1, a, b], c]

Out[4]=

PiecewiseExpand flattens it into a single Piecewise object.

In[5]:= PiecewiseExpand[%]

Out[5]=

Functions like Max and Abs, as well as Clip and UnitStep, implicitly involve conditionals, and combinations of them can again be reduced to a single Piecewise object using PiecewiseExpand.

This gives a result as a single Piecewise object.

In[6]:= PiecewiseExpand[Max[Min[a, b], c]]

Out[6]=

With x assumed real, this can also be written as a Piecewise object.

In[7]:= PiecewiseExpand[Abs[x], x \[Element] Reals]

Out[7]=

Functions like Floor, Mod and FractionalPart can also be expressed in terms of Piecewise objects, though in principle they can involve an infinite number of cases.

Without a bound on x, this would yield an infinite number of cases.

In[8]:= PiecewiseExpand[Floor[x^2], 0 < x < 2]

Out[8]=

Mathematica by default limits the number of cases that Mathematica will explicitly generate in the expansion of any single piecewise function such as Floor at any stage in a computation. You can change this limit by resetting the value of $MaxPiecewiseCases.