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 testi in turn, giving the value associated with the first one that is True|
|compare expr with each of the formi, giving the value associated with the first form it matches|
|use def as a default value|
|give the value corresponding to the first testi which yields True|
The test gives False
, so the "else
" expression y
Only the "else
" expression is evaluated in this case.
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
, and -1
This defines the positive part of the step function using a /;
Here is the negative part of the step function.
This shows the complete definition using /;
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
This defines a function with three regions. Using True
as the third test makes this the default case.
This uses the first case in the Which
This uses the third case.
This defines a function that depends on the values of its argument modulo 3
, so this uses the second case in the Switch
matches neither 0
, but does match _
An important point about symbolic systems such as Mathematica
is that the conditions you give may yield neither True
. Thus, for example, the condition xy
does not yield True
have specific values, such as numerical ones.
In this case, the test gives neither True
, so both branches in the If
You can add a special fourth argument to If
, which is used if the test does not yield True
|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.
leaves this as a symbolic equation.
is manifestly True
effectively assumes that expr
tests whether two expressions are manifestly identical. In this case, they are not.
The main difference between lhsrhs
always returns True
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.
operator gives a less useful result.
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
of the testi
always evaluates the testi
in turn, stopping if any of the testi
|expr1&&expr2&&expr3||evaluate until one of the expri is found to be False|
|expr1||expr2||expr3||evaluate until one of the expri is found to be True|
Evaluation of logical expressions.
This function involves a combination of two tests.
Here both tests are evaluated.
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.
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