LessEqual

x<=y or xy

yields True if x is determined to be less than or equal to y.

x1x2x3

yields True if the form a nondecreasing sequence.

Details

  • xy can be entered as x <= y or x \[LessEqual]y .
  • LessEqual gives True or False when its arguments are real numbers.
  • LessEqual does some simplification when its arguments are not numbers.
  • For exact numeric quantities, LessEqual internally uses numerical approximations to establish numerical ordering. This process can be affected by the setting of the global variable $MaxExtraPrecision.
  • In StandardForm, LessEqual is printed using .
  • xy, entered as x </ y or x \[LessSlantEqual] y, can be used on input as an alternative to xy.

Examples

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Basic Examples  (2)

Compare numbers:

Represent an inequality:

Scope  (9)

Numeric Inequalities  (7)

Inequalities are defined only for real numbers:

Compare rational numbers:

Approximate numbers that differ in at most their last eight binary digits are considered equal:

Compare an exact numeric expression and an approximate number:

Compare two exact numeric expressions; a numeric test may suffice to prove inequality:

Proving this inequality requires symbolic methods:

Symbolic and numeric methods used by LessEqual are insufficient to prove this inequality:

Use RootReduce to decide the sign of algebraic numbers:

Numeric methods used by LessEqual do not use sufficient precision to disprove this inequality:

RootReduce disproves the inequality using exact methods:

Increasing $MaxExtraPrecision may also disprove the inequality:

Symbolic Inequalities  (2)

Symbolic inequalities remain unevaluated, since x may not be a real number:

Use Refine to reevaluate the inequality assuming that x is real:

A symbolic inequality:

Use Reduce to find an explicit description of the solution set:

Use FindInstance to find a solution instance:

Use Minimize to optimize over the inequality-defined region:

Use Refine to simplify under the inequality-defined assumptions:

Properties & Relations  (12)

The negation of two-argument LessEqual is Greater:

The negation of three-argument LessEqual does not simplify automatically:

Use LogicalExpand to express it in terms of two-argument Greater:

This is not equivalent to three-argument Greater:

When LessEqual cannot decide inequality between numeric expressions, it returns unchanged:

FullSimplify uses exact symbolic transformations to prove the inequality:

NonPositive[x] is equivalent to :

Use Reduce to solve inequalities:

Use FindInstance to find solution instances:

Use RegionPlot and RegionPlot3D to visualize solution sets of inequalities:

Inequality assumptions:

Use Minimize and Maximize to solve optimization problems constrained by inequalities:

Use NMinimize and NMaximize to numerically solve constrained optimization problems:

Integrate a function over the solution set of inequalities:

Use Median, Quantile, and Quartiles to the ^(th) greatest number(s):

Possible Issues  (3)

Inequalities for machine-precision approximate numbers can be subtle:

The result is based on extra digits:

Arbitrary-precision approximate numbers do not have this problem:

Thanks to automatic precision tracking, LessEqual knows to look only at the first 10 digits:

In this case, inequality between machine numbers gives the expected result:

The extra digits in this case are ignored by LessEqual:

Introduced in 1988
 (1.0)
 |
Updated in 1996
 (3.0)