2.6.8 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[, , , ... ]	evaluate the in turn, giving the value associated with the first one that is True
Switch[expr, , , , ... ]
	compare expr with each of the , giving the value associated with the first form it matches
Switch[expr, , , , ... , _, def]
	use def as a default value
Piecewise[{{, }, ... }, def]
	give the value corresponding to the first which yields True

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

In[2]:=  If[7 > 8, Print[x], Print[y]]

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 shows the complete definition using /; conditions.

In[6]:=  ?g

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.

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.

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
lhs = rhs or SameQ[lhs, rhs]	give True if lhs and rhs are identical, and False otherwise
lhs = rhs 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

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

In[17]:=  x = y 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 Section 1.5.5. 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]:=  Head[a + b + c] = Times Out[18]=

The operator gives a less useful result.

In[19]:=  Head[a + b + c] Times Out[19]=

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

&& &&	evaluate until one of the is found to be False
|| ||	evaluate until one of the is found to be True

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.