How to | Create and Use Rules

Transformation rules in the Wolfram Language 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 Wolfram System 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 "x goes to 3":

By looking at the output for x3, 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 expr/.rules.

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, the Wolfram Language will use only the first rule:

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

Substitute 3y for x:

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 3 for x:

Use a rule for f[x]. Note that this rule matches f[x] exactly and does not affect f[y]:

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

The rule f[x_]x^2 can be read as "f[anything] goes to anything^2":

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 Wolfram System 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 x to 3, and then evaluate x to see the value:

Use a rule to assign a value for y:

Evaluating y, 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 p2 to n, and then use n with an expression:

Since p2 is now stored globally as the symbol n, you can continue to use n in place of p2.

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.

  • Applying Transformation Rules
  • Manipulating Sets of Transformation Rules
  • Transformation Rules for Functions
  • Introduction to Patterns
  • Patterns and Transformation Rules
  • Transformation Rules and Definitions
  • How to: Work with Variables and Functions
  • How to: Work with Rules
  • Rule  RuleDelayed  ReplaceAll  Set

  • Rules
  • Assignments
  • Options Management
  • Patterns
  • "How to" Topics