## Main Features

### A Clear Inheritance Structure

```
# Sets require elements to be unique and order doesn't matter
Set$new(1, 2, 1) == Set$new(1, 2)
#> [1] TRUE
Set$new(1, 2) == Set$new(2, 1)
#> [1] TRUE
# But tuples can enforce these restrictions
Tuple$new(1, 2, 1) != Tuple$new(1, 2)
#> [1] TRUE
Tuple$new(1, 2) != Tuple$new(2, 1)
#> [1] TRUE
# Fuzzy sets and tuples extend sets further
f = FuzzySet$new(1, 0, 2, 0.6, 3, 1)
f$inclusion(1)
#> [1] "Not Included"
f$inclusion(2)
#> [1] "Partially Included"
f$inclusion(3)
#> [1] "Fully Included"
# Symbolic intervals provide a clean way to represent infinite sets
Interval$new()
#> [-∞,+∞]
# Different closure types and classes are possible
Interval$new(1, 7, type = "(]") # half-open
#> (1,7]
Interval$new(1, 7, class = "integer") == Set$new(1:7)
#> [1] TRUE
# And SpecialSets inheriting from these intervals
Reals$new()
#> ℝ
PosRationals$new()
#> ℚ+
```

### Set operations

```
# Union
Set$new(1, 4, "a", "b") + Set$new(5)
#> {1, 4,...,a, b}
Interval$new(1, 5) + FuzzyTuple$new(1, 0.6)
#> [1,5]
# Power
Set$new(1:5)^2
#> {1, 2,...,4, 5}^2
# A symbolic representation is also possible
setpower(Set$new(1:5), power = 2, simplify = FALSE)
#> {1, 2,...,4, 5}^2
Reals$new()^5
#> ℝ^5
# Product
Set$new(1,2) * Set$new(5, 6)
#> {1, 2} × {5, 6}
Interval$new(1,5) * Tuple$new(3)
#> [1,5] × (3)
# Intersection
Set$new(1:5) & Set$new(4:10)
#> {4, 5}
ConditionalSet$new(function(x) x == 0) & Set$new(-2:2)
#> {0}
Interval$new(1, 10) & Set$new(5:6)
#> {5, 6}
# Difference
Interval$new(1, 10) - Set$new(5)
#> [1,5) ∪ (5,10]
Set$new(1:5) - Set$new(2:3)
#> {1, 4, 5}
```

### Containedness and Comparators

```
Interval$new(1, 10)$contains(5)
#> [1] TRUE
# check multiple elements
Interval$new(1, 10)$contains(8:12)
#> [1] TRUE TRUE TRUE FALSE FALSE
# only return TRUE if all are TRUE
Interval$new(1, 10)$contains(8:12, all = TRUE)
#> [1] FALSE
# decide whether open bounds should be included
Interval$new(1, 10, type = "()")$contains(10, bound = TRUE)
#> [1] TRUE
Interval$new(1, 10, type = "()")$contains(10, bound = TRUE)
#> [1] TRUE
Interval$new(1, 5, class = "numeric")$equals(Set$new(1:5))
#> [1] FALSE
Interval$new(1, 5, class = "integer")$equals(Set$new(1:5))
#> [1] TRUE
Set$new(1) == FuzzySet$new(1, 1)
#> [1] TRUE
# proper subsets
Set$new(1:3)$isSubset(Set$new(1), proper = TRUE)
#> [1] TRUE
Set$new(1) < Set$new(1:3)
#> [1] TRUE
# (non-proper) subsets
Set$new(1:3)$isSubset(Set$new(1:3), proper = FALSE)
#> [1] TRUE
Set$new(1:3) <= Set$new(1:3)
#> [1] TRUE
# multi-dimensional checks
x = PosReals$new()^2
x$contains(list(Tuple$new(1, 1), Tuple$new(-2, 3)))
#> [1] TRUE FALSE
```