Returns the symmetric difference of two objects inheriting from class Set.

setsymdiff(x, y, simplify = TRUE)

x %-% y

Arguments

x, y

Set

simplify

logical, if TRUE (default) returns the result in its simplest form, usually a Set or UnionSet, otherwise a ComplementSet.

Value

An object inheriting from Set containing the symmetric difference of elements in both x and y.

Details

The symmetric difference, aka disjunctive union, of two sets, \(X, Y\), is defined as the set of elements that exist in set \(X\) or in \(Y\) but not both, $$\{z : (z \epsilon X \cup z \epsilon Y) \\ \cap \\ \neg(z \epsilon X \cap z \epsilon Y)\}$$

The symmetric difference can also be expressed as the union of two sets minus the intersection. Therefore setsymdiff is written as a thin wrapper over these operations, so for two sets, A,B:
A %-% B = (A | B) - (A & B).

The symmetric difference of fuzzy and crisp sets first coerces fuzzy sets to crisp sets by finding their support.

See also

Examples

# symmetrical difference compared to union and intersection Set$new(1, 2, 3) %-% Set$new(3, 4)
#> {1, 2, 4}
(Set$new(1, 2, 3) | Set$new(3, 4)) - (Set$new(1, 2, 3) & Set$new(3, 4))
#> {1, 2, 4}
# ConditionalSets demonstrate the internal logic ConditionalSet$new(function(x) x > 0) %-% ConditionalSet$new(function(y) y == 0)
#> isSubset is currently undefined for conditional sets.
#> isSubset is currently undefined for conditional sets.
#> {(x > 0 | y == 0) & !(x > 0 & y == 0) : x ∈ V, y ∈ V}