How to | Create and Use Rules
Transformation rules in Mathematica let you set local values for symbols, functions, and all other types of expressions. Using rules provides a powerful and extensible method to replace all or part of another expression with the value you specify.
The short form for a rule uses a right arrow, which you get by typing (with no space between - and >). The Mathematica front end automatically converts into upon further typing. Either symbol is a short form for Rule.
Create the following transformation rule, which can be thought of as " goes to ":
By looking at the output for , you can see that this rule does not do anything: the output is simply the rule itself. This is because rules do not do anything when they are alone. You must use a rule with an expression for it to be of any use.
Rules can be applied to expressions by using (the short form for ReplaceAll). The general syntax for this is .
Use to use a rule with an expression:
To use two or more rules with an expression, place them in a list :
If you give two rules for the same variable, Mathematica will use only the first rule:
You can replace variables with any expression, not just individual values.
Substitute for :
You can also use a rule to replace larger parts of an expression:
In fact, you can use rules with any expression, including functions.
Substitute for :
Use a rule for . Note that this rule matches exactly and does not affect :
To replace the function regardless of its argument, you must use a pattern in the rule.
The rule can be read as " goes to ":
For more information on using patterns, see "Introduction to Patterns."
Rules that are set up using are immediate rules. That is, the right-hand side is evaluated at the same time as the rule:
You may need to use delayed rules instead, which are not evaluated until they are used with an expression. Delayed rules are created by using RuleDelayed.
The short form for a delayed rule is (with no space between : and >). The Mathematica front end automatically converts into upon typing. Either represents the short form for RuleDelayed:
Consider a problem where you want to use a rule to generate three random real numbers in the range of 0 to 1. Using an immediate rule results in the generation of the same three numbers:
To generate three different numbers, use a delayed rule:
Assignments set explicitly using have a global effect, while rules only affect the expression with which they are used.
Use to assign to , and then evaluate to see the value:
Use a rule to assign a value for :
Evaluating , you can see that the value assigned by the rule was not saved:
You must use a rule with an expression for it to work. However, you can explicitly assign a rule to a symbol and then use that symbol as you would the rule.
Use to assign the rule to , and then use with an expression:
Since is now stored globally as the symbol , you can continue to use in place of .
Similarly, you can explicitly assign an expression to a symbol and then use a rule on the symbol:
This is especially convenient if you plan to use the expression in more than one computation.