## Documentation |

Inequalities "less than or equal to" and "greater than or equal to"

This functionality does not run in MATLAB.

x <= yx >= y_leequal(x,y)

`x <= y` and `x >= y` define
inequalities.

`x <= y` represents the Boolean statement
"`x` is less than or equal to `y`".
It is equivalent to the function call `_leequal(x,y)`.

`x >= y` represents the Boolean statement
"`x` is greater than or equal to `y`".
It is always converted to `y <= x`, which is equivalent
to the function call `_leequal(y,x)`.

These operators return symbolic Boolean expressions. If only
real numbers of `Type::Real` are
involved, these expressions can be evaluated to `TRUE` or `FALSE` by
the function `bool`.
They also serve as control conditions in `if`, `repeat`, and `while` statements. For floating-point intervals, these operators
are interpreted as "strictly smaller than" and so on,
see Example 2.

Further, Boolean expressions can be evaluated to `TRUE`, `FALSE`,
or `UNKNOWN` by
the function `is`.
Tests using `is` can
also be applied to constant symbolic expressions. See Example 4.

`bool` also
handles inequalities involving character strings. It compares them
with respect to the lexicographical ordering.

Inequalities have two operands: the left hand side and the right
hand side. One may use `lhs` and `rhs` to extract these
operands.

The operators `<`, `<=`, `>`,
and `>=` produce symbolic inequalities. They can
be evaluated to `TRUE` or `FALSE` by
the function `bool` if
only real numbers of type `Type::Real` (integers,
rationals, and floats) are involved:

1.5 <= 3/2; bool(%)

Note that `bool` may
fail to handle Boolean expressions that involve exact expressions,
even if they represent real numbers:

_less(PI, sqrt(2) + 17/10); bool(%)

bool(sqrt(6) < sqrt(2)*sqrt(3))

Error: Cannot evaluate to Boolean. [_less]

Comparison of intervals is interpreted as "strict", that is, all combinations of numbers in the intervals must fulfill the relation:

bool(0...1 < 2...3), bool(0...2 < 1...3), bool(0...1 < 1...2)

bool(0...1 <= 2...3), bool(0...2 <= 1...3), bool(0...1 <= 1...2)

This examples demonstrates how character strings can be compared:

if "text" < "t"."e"."x"."t"."book" then "yes" else "no" end

bool("a" >= "b")

Note that `bool` does not perform symbolic
simplification and therefore cannot handle some combinations of symbolic
expressions; the function `is` does
perform symbolic simplification:

bool(sqrt(6) < sqrt(2)*sqrt(3))

Error: Cannot evaluate to Boolean. [_less]

is(sqrt(6) < sqrt(2)*sqrt(3))

Inequalities are valid input objects for the system function `solve`:

solve(x^2 - 2*x < 3, x)

solve(x^2 - 2*x >= 3, x)

The operators `<` and `<=` can
be overloaded by
user-defined domains:

myDom := newDomain("myDom"): myDom::print := x -> extop(x):

Without overloading `_less` or `_leequal`,
elements of this domain cannot be compared:

x := new(myDom, PI): y := new(myDom, sqrt(10)): bool(x < y)

Error: Cannot evaluate to Boolean. [_less]

Now, a slot`"_less"` is
defined. It is called, when an inequality of type `"_less"` is
evaluated by `bool`.
The slot compares floating-point approximations if the arguments are
not of type `Type::Real`:

myDom::_less := proc(x, y) begin x := extop(x, 1): y := extop(y, 1): if not testtype(x, Type::Real) then x := float(x): if not testtype(x, Type::Real) then error("cannot compare") end_if end_if: if not testtype(y, Type::Real) then y := float(y): if not testtype(y, Type::Real) then error("cannot compare") end_if end_if: bool(x < y) end_proc:

x, y, bool(x < y), bool(x > y)

bool(new(myDom, I) < new(myDom, PI))

Error: cannot compare [myDom::_less]

delete myDom, x, y:

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