x<=y or x≤y
yields True if x is determined to be less than or equal to y.
yields True if the form a nondecreasing sequence.
- LessEqual is also known as weak inequality or non-strict inequality.
- x≤y 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 ≤.
- x≤y, entered as x </ y or x \[LessSlantEqual] y, can be used on input as an alternative to x≤y.
Examplesopen allclose all
Numeric Inequalities (7)
Inequalities are defined only for real 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:
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:
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 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:
Wolfram Research (1988), LessEqual, Wolfram Language function, https://reference.wolfram.com/language/ref/LessEqual.html (updated 1996).
Wolfram Language. 1988. "LessEqual." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. https://reference.wolfram.com/language/ref/LessEqual.html.
Wolfram Language. (1988). LessEqual. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LessEqual.html