R: Intervals

interval {sets} | R Documentation

## Intervals

### Description

Interval class for countable and uncountable numeric sets.

### Usage

    interval(l=NULL, r=l,
             bounds=c("[]", "[)", "(]", "()", "[[", "]]", "][",
                      "open", "closed", "left-open", "right-open",
                      "left-closed", "right-closed"),
             domain=NULL)

    reals(l=NULL, r=NULL,
          bounds=c("[]", "[)", "(]", "()", "[[", "]]", "][",
                   "open", "closed", "left-open", "right-open",
                   "left-closed", "right-closed"))
    integers(l=NULL, r=NULL)
    naturals(l=NULL, r=NULL)
    naturals0(l=NULL, r=NULL)
    l %..% r

    interval_domain(x)

    as.interval(x)
    integers2reals(x, min=-Inf, max=Inf)
    reals2integers(x)

    interval_complement(x, y=NULL)
    interval_intersection(...)
    interval_symdiff(...)
    interval_union(...)

    interval_difference(...)
    interval_division(...)
    interval_product(...)
    interval_sum(...)

    is.interval(x)
    interval_contains_element(x, y)
    interval_is_bounded(x)
    interval_is_closed(x)
    interval_is_countable(...)
    interval_is_degenerate(x)
    interval_is_empty(x)
    interval_is_equal(x, y)
    interval_is_less_than_or_equal(x, y)
    interval_is_less_than(x, y)
    interval_is_greater_than_or_equal(x, y)
    interval_is_greater_than(x, y)
    interval_is_finite(x)
    interval_is_half_bounded(x)
    interval_is_left_bounded(x)
    interval_is_left_closed(x)
    interval_is_left_open(...)
    interval_is_left_unbounded(x)
    interval_measure(x)
    interval_is_proper(...)
    interval_is_proper_subinterval(x, y)
    interval_is_right_bounded(x)
    interval_is_right_closed(x)
    interval_is_right_open(...)
    interval_is_right_unbounded(x)
    interval_is_subinterval(x, y)
    interval_is_unbounded(x)
    interval_is_uncountable(x)
    interval_power(x, n)
    x %<% y
    x %>% y
    x %<=% y
    x %>=% y

### Arguments

`x` | For `as.interval()` and `is.interval()`: an R object. For all other functions: an interval object (or any other R object coercible to one).
`y` | An interval object (or any other R object coercible to one).
`min, max` | Integers defining the range to be coerced.
`l, r` | Numeric values defining the bounds of the interval. For integer domains, these will be rounded.
`bounds` | Character string specifying whether the interval is open, closed, or left/right-open/closed. Symbolic shortcuts such as `"()"` or `"]["` for an open interval, etc., are also accepted.
`domain` | Character string specifying the domain of the interval: `"R"`, `"Z"`, `"N"`, and `"N0"` for the reals, integers, positive integers and non-negative integers, respectively. If unspecified, the domain will be guessed from the mode of the numeric values specifying the bounds.
`n` | Integer exponent.
`...` | Interval objects (or other R objects coercible to interval objects).

### Details

An interval object represents a multi-interval, i.e., a union of disjoint, possibly unbounded (i.e., infinite) ranges of numbers—either the extended reals, or sequences of integers. The usual set operations (union, complement, intersection) and predicates (equality, (proper) inclusion) are implemented. If (numeric) sets and interval objects are mixed, the result will be an interval object. Some basic interval arithmetic operations (addition, subtraction, multiplication, division, power) as well mathematical functions (`log`, `log2`, `log10`, `exp`, `abs`, `sqrt`, `trunc`, `round`, `floor`, `ceiling`, `signif`, and the trigonometric functions) are defined. Note that the rounding functions will discretize the interval.

Coercion methods for the `as.numeric`, `as.list`, and `as.set` generics are implemented. `reals2integers()` discretizes a real multi-interval. `integers2reals()` returns a multi-interval of corresponding (degenerate) real intervals.

The summary functions `min`, `max`, `range`, `sum`, `mean` and `prod` are implemented and work on the interval bounds.

`sets_options()` allows to change the style of open bounds according to the ISO 31-11 standard using reversed brackets instead of round parentheses (see examples).

### Value

For the predicates: a logical value. For all other functions: an interval object.

### See Also

`set` and `gset` for finite (generalized) sets.

### Examples

    #### * general interval constructor

    interval(1,5)
    interval(1,5, "[)")
    interval(1,5, "()")

    ## ambiguous notation -> use alternative style
    sets_options("openbounds", "][")
    interval(1,5, "()")
    sets_options("openbounds", "()")

    interval(1,5, domain = "Z")
    interval(1L, 5L)

    ## degenerate interval
    interval(3)

    ## empty interval
    interval()

    #### * reals
    reals()
    reals(1,5)
    reals(1,5,"()")
    reals(1) ## half-unbounded

    ## (auto-)complement
    !reals(1,5)
    interval_complement(reals(1,5), reals(2, Inf))

    ## combine/c(reals(2,4), reals(3,5))
    reals(2,4) | reals(3,5)

    ## intersection
    reals(2,4) & reals(3,5)

    ## overlapping intervals
    reals(2,4) & reals(3,5)
    reals(2,4) & reals(4,5,"(]")

    ## non-overlapping
    reals(2,4) & reals(7,8)
    reals(2,4) | reals(7,8)
    reals(2,4,"[)") | reals(4,5,"(]")

    ## degenerated cases
    reals(2,4) | interval()
    c(reals(2,4), set())

    reals(2,4) | interval(6)
    c(reals(2,4), set(6), 9)

    ## predicates
    interval_is_empty(interval())
    interval_is_degenerate(interval(4))
    interval_is_bounded(reals(1,2))
    interval_is_bounded(reals(1,Inf)) ## !! FALSE, because extended reals
    interval_is_half_bounded(reals(1,Inf))
    interval_is_left_bounded(reals(1,Inf))
    interval_is_right_unbounded(reals(1,Inf))
    interval_is_left_closed(reals(1,Inf))
    interval_is_right_closed(reals(1,Inf)) ## !! TRUE

    reals(1,2) <= reals(1,5)
    reals(1,2) < reals(1,2)
    reals(1,2) <= reals(1,2,"[)")
    reals(1,2,"[)") < reals(1,2)

    #### * integers
    integers()
    naturals()
    naturals0()

    3 %..% 5
    integers(3, 5)
    integers(3, 5) | integers(6,9)
    integers(3, 5) | integers(7,9)

    interval_complement(naturals(), integers())

    naturals() <= naturals0()
    naturals0() <= integers()

    ## mix reals and integers
    c(reals(2,5), integers(7,9))
    interval_complement(reals(2,5), integers())
    interval_complement(integers(2,5), reals())

    try(interval_complement(integers(), reals()), silent = TRUE)
    ## infeasible --> error

    integers() <= reals()
    reals() <= integers()

    ### interval arithmetic
    x <- interval(2,4)
    y <- interval(3,6)
    x + y
    x - y
    x * y
    x / y

    ## summary functions
    min(x, y)
    max(y)
    range(y)
    mean(y)

[Package sets version 1.0-18 Index]