MATHEMATICA HOW TO

# 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 ":

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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:

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To use two or more rules with an expression, place them in a list :

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If you give two rules for the same variable, Mathematica will use only the first rule:

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You can replace variables with any expression, not just individual values.

Substitute for :

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You can also use a rule to replace larger parts of an expression:

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In fact, you can use rules with any expression, including functions.

Substitute for :

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Use a rule for . Note that this rule matches exactly and does not affect :

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To replace the function regardless of its argument, you must use a pattern in the rule.

The rule can be read as " goes to ":

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Rules that are set up using are immediate rules. That is, the right-hand side is evaluated at the same time as the rule:

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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:

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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:

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To generate three different numbers, use a delayed rule:

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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:

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Use a rule to assign a value for :

Evaluating , you can see that the value assigned by the rule was not saved:

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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:

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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:

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This is especially convenient if you plan to use the expression in more than one computation.