Mathematica Tutorial

Conditionals

Mathematica provides various ways to set up conditionals, which specify that particular expressions should be evaluated only if certain conditions hold.

lhs:=rhs/;test	use the definition only if test evaluates to True
If[test,then,else]	evaluate then if test is True, and else if it is False
Which[test₁,value₁,test₂,...]	evaluate the test_i in turn, giving the value associated with the first one that is True
Switch[expr,form₁,value₁,form₂,...]
	compare expr with each of the form_i, giving the value associated with the first form it matches
Switch[expr,form₁,value₁,form₂,...,_,def]
	use def as a default value
Piecewise[{{value₁,test₁},...},def]
	give the value corresponding to the first test_i which yields True

Conditional constructs.

The test gives False, so the "else" expression y is returned.

In[1]:=

Out[1]=

Only the "else" expression is evaluated in this case.

In[2]:=

When you write programs in Mathematica, you will often have a choice between making a single definition whose right-hand side involves several branches controlled by If functions, or making several definitions, each controlled by an appropriate /; condition. By using several definitions, you can often produce programs that are both clearer, and easier to modify.

This defines a step function, with value 1 for x>0, and -1 otherwise.

In[3]:=

This defines the positive part of the step function using a /; condition.

In[4]:=

Here is the negative part of the step function.

In[5]:=

This shows the complete definition using /; conditions.

In[6]:=

The function If provides a way to choose between two alternatives. Often, however, there will be more than two alternatives. One way to handle this is to use a nested set of If functions. Usually, however, it is instead better to use functions like Which and Switch.

This defines a function with three regions. Using True as the third test makes this the default case.

In[7]:=

This uses the first case in the Which.

In[8]:=

Out[8]=

This uses the third case.

In[9]:=

Out[9]=

This defines a function that depends on the values of its argument modulo 3.

In[10]:=

Mod[7, 3] is 1, so this uses the second case in the Switch.

In[11]:=

Out[11]=

17 matches neither 0 nor 1, but does match _.

In[12]:=

Out[12]=

An important point about symbolic systems such as Mathematica is that the conditions you give may yield neither True nor False. Thus, for example, the condition x

y does not yield True or False unless x and y have specific values, such as numerical ones.

In this case, the test gives neither True nor False, so both branches in the If remain unevaluated.

In[13]:=

Out[13]=

You can add a special fourth argument to If, which is used if the test does not yield True or False.

In[14]:=

Out[14]=

If[test,then,else,unknown]	a form of If which includes the expression to use if test is neither True nor False
TrueQ[expr]	give True if expr is True, and False otherwise
lhsrhs or SameQ[lhs,rhs]	give True if lhs and rhs are identical, and False otherwise
lhsrhs or UnsameQ[lhs,rhs]	give True if lhs and rhs are not identical, and False otherwise
MatchQ[expr,form]	give True if the pattern form matches expr, and give False otherwise

Functions for dealing with symbolic conditions.

Mathematica leaves this as a symbolic equation.

In[15]:=

Out[15]=

Unless expr is manifestly True, TrueQ[expr] effectively assumes that expr is False.

In[16]:=

Out[16]=

Unlike

tests whether two expressions are manifestly identical. In this case, they are not.

In[17]:=

Out[17]=

The main difference between lhs

rhs and lhs

rhs is that

always returns True or False, whereas

can leave its input in symbolic form, representing a symbolic equation, as discussed in "Equations". You should typically use

when you want to test the structure of an expression, and

if you want to test mathematical equality. The Mathematica pattern matcher effectively uses

to determine when one literal expression matches another.

You can use

to test the structure of expressions.

In[18]:=

Out[18]=

The

operator gives a less useful result.

In[19]:=

Out[19]=

In setting up conditionals, you will often need to use combinations of tests, such as test₁&&test₂&&.... An important point is that the result from this combination of tests will be False if any of the test_i yield False. Mathematica always evaluates the test_i in turn, stopping if any of the test_i yield False.

expr₁&&expr₂&&expr₃	evaluate until one of the expr_i is found to be False
expr₁\|\|expr₂\|\|expr₃	evaluate until one of the expr_i is found to be True

Evaluation of logical expressions.

This function involves a combination of two tests.

In[20]:=

Here both tests are evaluated.

In[21]:=

Out[21]=

Here the first test yields False, so the second test is not tried. The second test would involve 1/0, and would generate an error.

In[22]:=

Out[22]=

The way that Mathematica evaluates logical expressions allows you to combine sequences of tests where later tests may make sense only if the earlier ones are satisfied. The behavior, which is analogous to that found in languages such as C, is convenient in constructing many kinds of Mathematica programs.