diff --git a/src/BangBang.jl b/src/BangBang.jl
index 4c4b20e..723b40a 100644
--- a/src/BangBang.jl
+++ b/src/BangBang.jl
@@ -14,6 +14,7 @@ export @!,
        deleteat!!,
        empty!!,
        finish!,
+       intersect!!,
        lmul!!,
        materialize!!,
        merge!!,
@@ -31,6 +32,7 @@ export @!,
        setproperty!!,
        singletonof,
        splice!!,
+       symdiff!!,
        union!!,
        unique!!
 
@@ -58,6 +60,8 @@ include("utils.jl")
 function implements end
 function push!! end
 function unique!! end
+function union!! end
+function symdiff!! end
 
 include("NoBang/NoBang.jl")
 using .NoBang: Empty, SingletonVector, singletonof
diff --git a/src/NoBang/NoBang.jl b/src/NoBang/NoBang.jl
index 0914890..47948c3 100644
--- a/src/NoBang/NoBang.jl
+++ b/src/NoBang/NoBang.jl
@@ -7,7 +7,7 @@ using Base: ImmutableDict
 using Requires
 using ConstructionBase: constructorof, setproperties
 
-using ..BangBang: push!!, unique!!, implements
+using ..BangBang: push!!, unique!!, union!!, symdiff!!, implements
 
 include("singletoncontainers.jl")
 include("base.jl")
diff --git a/src/NoBang/base.jl b/src/NoBang/base.jl
index fbcb5e0..7b911de 100644
--- a/src/NoBang/base.jl
+++ b/src/NoBang/base.jl
@@ -151,6 +151,10 @@ setproperty(value, name, x) = setproperties(value, NamedTuple{(name,)}((x,)))
 
 materialize(::Any, x) = Broadcast.materialize(x)
 
-@inline _union(a, b) = union(a, b)
+@inline _union(args...) = union(args...)
 
-@inline _setdiff(a, b) = setdiff(a, b)
+@inline _intersect(args...) = intersect(args...)
+
+@inline _setdiff(args...) = setdiff(args...)
+
+@inline _symdiff(args...) = symdiff(args...)
diff --git a/src/NoBang/emptycontainers.jl b/src/NoBang/emptycontainers.jl
index c165457..ed7ee87 100644
--- a/src/NoBang/emptycontainers.jl
+++ b/src/NoBang/emptycontainers.jl
@@ -75,10 +75,15 @@ append(e::Empty, ::Empty) = e
 _empty(x::Empty) = x
 resize(::Empty{T}, n::Integer) where {T <: AbstractVector} = similar(T, (n,))
 
-_union(::Empty{T}, x) where {T} = unique!!(T(x))
-_union(e::Empty, ::Empty) = e
+_union(::Empty{T}, x, ys...) where {T} = union!!(unique!!(T(x)), ys...)
+_union(e::Empty, ::Empty...) = e
 
-_setdiff(x::Empty, _) = x
+_intersect(x::Empty, ys...) = x
+
+_setdiff(x::Empty, ys...) = x
+
+_symdiff(::Empty{T}, x, ys...) where {T} = symdiff!!(unique!!(T(x)), ys...)
+_symdiff(e::Empty, ::Empty...) = e
 
 _setindex(::Empty{T}, v, k) where {T <: AbstractDict} = T(SingletonDict(k, v))
 Base.get(::Empty, _, default) = default
diff --git a/src/base.jl b/src/base.jl
index bace124..be631e5 100644
--- a/src/base.jl
+++ b/src/base.jl
@@ -710,15 +710,50 @@ false
 ```
 """
 union!!
-union!!(set, itr) = may(union!, set, itr)
-union!!(set, itr, itrs...) = foldl(union!!, itrs, init = union!!(set, itr))
+union!!(args...) = may(union!, args...)
 
 pure(::typeof(union!)) = NoBang._union
 _asbb(::typeof(union!)) = union!!
-possible(::typeof(union!), ::C, ::I) where {C<:Union{AbstractSet,AbstractVector},I} =
+possible(::typeof(union!), ::C, ts...) where C<:Union{AbstractSet,AbstractVector} =
     implements(push!, C) &&
-    IteratorEltype(I) isa HasEltype && promote_type(eltype(C), eltype(I)) <: eltype(C)
-possible(::typeof(union!), ::Empty, ::Any) = false
+    all(t -> IteratorEltype(t) isa HasEltype, ts) &&
+    promote_type(eltype(C), map(eltype, ts)...) <: eltype(C)
+possible(::typeof(union!), ::Empty, ts...) = false
+# TODO: simplify `possible` after deciding what to do with dictionaries
+
+"""
+    intersect!!(setlike, others...) -> setlike′
+
+Return the set of elements in `setlike` and in all the collections
+`others`. Mutate `setlike` if possible.
+
+This function "owns" `setlike` but not `others`; i.e., returned value
+`setlike′` does not alias any of `others`.  For example,
+`other = Set([1]); intersect!!(Set([1, 2]), other)` does not return `other` as-is.
+
+# Examples
+```jldoctest
+julia> using BangBang
+
+julia> xs = Set([1, 2]);
+
+julia> ys = intersect!!(xs, [1]);  # mutates `xs` as it's possible
+
+julia> ys == Set([1])
+true
+
+julia> ys === xs  # `xs` is returned
+true
+```
+"""
+intersect!!
+intersect!!(args...) = may(intersect!, args...)
+
+pure(::typeof(intersect!)) = NoBang._intersect
+_asbb(::typeof(intersect!)) = intersect!!
+possible(::typeof(intersect!), ::C, ::Any...) where {C<:Union{AbstractSet,AbstractVector}} =
+    implements(push!, C)
+possible(::typeof(intersect!), ::Empty, ts...) = false
 # TODO: simplify `possible` after deciding what to do with dictionaries
 
 """
@@ -729,8 +764,7 @@ Return the set of elements in `setlike` but not in any of the collections
 
 This function "owns" `setlike` but not `others`; i.e., returned value
 `setlike′` does not alias any of `others`.  For example,
-`setdiff!!(Empty(Set), other)` shallow-copies `other` instead of
-returning `other` as-is.
+`other = Set([]); setdiff!!(Empty(Set), other)` does not return `other` as-is.
 
 # Examples
 ```jldoctest
@@ -748,12 +782,49 @@ true
 ```
 """
 setdiff!!
-setdiff!!(set, itr) = may(setdiff!, set, itr)
-setdiff!!(set, itr, itrs...) = foldl(setdiff!!, itrs, init = setdiff!!(set, itr))
+setdiff!!(args...) = may(setdiff!, args...)
 
 pure(::typeof(setdiff!)) = NoBang._setdiff
 _asbb(::typeof(setdiff!)) = setdiff!!
-possible(::typeof(setdiff!), ::C, ::Any) where {C<:Union{AbstractSet,AbstractVector}} =
+possible(::typeof(setdiff!), ::C, ::Any...) where {C<:Union{AbstractSet,AbstractVector}} =
     implements(push!, C)
-possible(::typeof(setdiff!), ::Empty, ::Any) = false
+possible(::typeof(setdiff!), ::Empty, ts...) = false
+# TODO: simplify `possible` after deciding what to do with dictionaries
+
+"""
+    symdiff!!(setlike, others...) -> setlike′
+
+Return the set of elements that occur an odd number of times in
+`setlike` and the `others` collections. Mutate `setlike` if possible.
+
+This function "owns" `setlike` but not `others`; i.e., returned value
+`setlike′` does not alias any of `others`.  For example,
+`symdiff!!(Empty(Set), other)` shallow-copies `other` instead of
+returning `other` as-is.
+
+# Examples
+```jldoctest
+julia> using BangBang
+
+julia> xs = Set([1, 2]);
+
+julia> ys = symdiff!!(xs, [2, 3]);  # mutates `xs` as it's possible
+
+julia> ys == Set([1, 3])
+true
+
+julia> ys === xs  # `xs` is returned
+true
+```
+"""
+symdiff!!
+symdiff!!(args...) = may(symdiff!, args...)
+
+pure(::typeof(symdiff!)) = NoBang._symdiff
+_asbb(::typeof(symdiff!)) = symdiff!!
+possible(::typeof(symdiff!), ::C, ts...) where C<:Union{AbstractSet,AbstractVector} =
+    implements(push!, C) &&
+    all(t -> IteratorEltype(t) isa HasEltype, ts) &&
+    promote_type(eltype(C), map(eltype, ts)...) <: eltype(C)
+possible(::typeof(symdiff!), ::Empty, ts...) = false
 # TODO: simplify `possible` after deciding what to do with dictionaries
diff --git a/src/initials.jl b/src/initials.jl
index 5b658f2..5d5425d 100644
--- a/src/initials.jl
+++ b/src/initials.jl
@@ -84,3 +84,15 @@ InitialValues.hasinitialvalue(::Type{typeof(union!!)}) = true
 
 copyunionable(src::AbstractVector) = Base.copymutable(src)
 copyunionable(src) = Set(src)
+
+const InitIntersect!! = InitialValues.GenericInitialValue{typeof(intersect!!)}
+intersect!!(dest::InitIntersect!!, src) = copyunionable(src)
+intersect!!(dest, ::InitIntersect!!) = dest
+intersect!!(dest::InitIntersect!!, src::InitIntersect!!) = dest
+InitialValues.hasinitialvalue(::Type{typeof(intersect!!)}) = true
+
+const InitSymdiff!! = InitialValues.GenericInitialValue{typeof(symdiff!!)}
+symdiff!!(dest::InitSymdiff!!, src) = copyunionable(src)
+symdiff!!(dest, ::InitSymdiff!!) = dest
+symdiff!!(dest::InitSymdiff!!, src::InitSymdiff!!) = dest
+InitialValues.hasinitialvalue(::Type{typeof(symdiff!!)}) = true
diff --git a/test/test_intersect.jl b/test/test_intersect.jl
new file mode 100644
index 0000000..56cf8c5
--- /dev/null
+++ b/test/test_intersect.jl
@@ -0,0 +1,25 @@
+module TestIntersect
+
+include("preamble.jl")
+# using MicroCollections: SingletonSet
+
+@testset begin
+    @test_returns_first intersect!!([0]) == [0]
+    @test_returns_first intersect!!(Set([0])) == Set([0])
+    @test_returns_first intersect!!([0], [1]) == []
+    @test_returns_first intersect!!([0], (1,)) == []
+    @test_returns_first intersect!!([0, 1, 2], (0, 1), [1, 2]) == [1]
+    @test_returns_first intersect!!(Set([0]), [0]) == Set([0])
+    @test_returns_first intersect!!(Set([0]), (0,)) == Set([0])
+    @test_returns_first intersect!!(Set([0, 1, 2]), Set([0,1]), [1, 2]) == Set([1])
+    @test intersect!!(Empty(Set)) === Empty(Set)
+    @test intersect!!(Empty(Vector), Empty(Set)) === Empty(Vector)
+    @test intersect!!(Empty(Vector), [0]) === Empty(Vector)
+    @test intersect!!(Empty(Set{Int}), [0]) === Empty(Set{Int})
+    #=
+    @test intersect!!(SingletonSet((0,)), [0]) ==ₜ Set([0])
+    @test intersect!!(SingletonSet((0,)), [1]) ==ₜ Set{Int}()
+    =#
+end
+
+end  # module
diff --git a/test/test_setdiff.jl b/test/test_setdiff.jl
index bdafd15..736abc2 100644
--- a/test/test_setdiff.jl
+++ b/test/test_setdiff.jl
@@ -4,13 +4,16 @@ include("preamble.jl")
 # using MicroCollections: SingletonSet
 
 @testset begin
+    @test_returns_first setdiff!!([0]) == [0]
+    @test_returns_first setdiff!!(Set([0])) == Set([0])
     @test_returns_first setdiff!!([0], [0]) == []
     @test_returns_first setdiff!!([0], (0,)) == []
     @test_returns_first setdiff!!([0, 1, 2], (0,), [1]) == [2]
     @test_returns_first setdiff!!(Set([0]), [0]) == Set()
     @test_returns_first setdiff!!(Set([0]), (0,)) == Set()
     @test_returns_first setdiff!!(Set([0, 1, 2]), Set([0]), [1]) == Set([2])
-    @test setdiff!!(Empty(Vector), []) === Empty(Vector)
+    @test setdiff!!(Empty(Set)) === Empty(Set)
+    @test setdiff!!(Empty(Vector), Empty(Set)) === Empty(Vector)
     @test setdiff!!(Empty(Set{Int}), []) === Empty(Set{Int})
     #=
     @test setdiff!!(SingletonSet((0,)), [0]) ==ₜ Set{Int}()
diff --git a/test/test_symdiff.jl b/test/test_symdiff.jl
new file mode 100644
index 0000000..51c79a4
--- /dev/null
+++ b/test/test_symdiff.jl
@@ -0,0 +1,24 @@
+module TestSymdiff
+
+include("preamble.jl")
+# using MicroCollections: SingletonSet
+
+@testset begin
+    @test_returns_first symdiff!!([0]) == [0]
+    @test_returns_first symdiff!!(Set([0])) == Set([0])
+    @test_returns_first symdiff!!([0], [0]) == []
+    @test_returns_first symdiff!!([0], (0,)) == []
+    @test_returns_first symdiff!!([0, 1, 2], (0, 1), [1, 3]) == [1, 2, 3]
+    @test_returns_first symdiff!!(Set([0]), [0]) == Set()
+    @test_returns_first symdiff!!(Set([0]), (1,)) == Set([0, 1])
+    @test_returns_first symdiff!!(Set([0, 1, 2]), Set([0, 1]), [1]) == Set([1, 2])
+    @test symdiff!!(Empty(Set)) === Empty(Set)
+    @test symdiff!!(Empty(Vector), Empty(Vector)) === Empty(Vector)
+    @test symdiff!!(Empty(Set{Int}), Empty(Vector)) === Empty(Set{Int})
+    #=
+    @test symdiff!!(SingletonSet((0,)), [0]) ==ₜ Set{Int}()
+    @test symdiff!!(SingletonSet((0,)), [1]) ==ₜ Set([0, 1])
+    =#
+end
+
+end  # module
diff --git a/test/test_union.jl b/test/test_union.jl
index 3dfc24f..e91d47c 100644
--- a/test/test_union.jl
+++ b/test/test_union.jl
@@ -5,6 +5,8 @@ using BangBang: SingletonVector
 using InitialValues: InitialValue, asmonoid
 
 @testset begin
+    @test_returns_first union!!([0]) == [0]
+    @test_returns_first union!!(Set([0])) == Set([0])
     @test union!!([0.0], [1.0]) ==ₜ [0.0, 1.0]
     @test union!!([0], [0.5]) ==ₜ [0.0, 0.5]
     @test union!!(Set([0.0]), [1.0]) ==ₜ Set([0.0, 1.0])
@@ -18,6 +20,7 @@ end
         @test union!!(Empty(Set), Empty(Set)) === Empty(Set)
         @test union!!(Empty(Set), Empty(Vector)) === Empty(Set)
     end
+    @test union!!(Empty(Set)) === Empty(Set)
 end
 
 @testset "mutation" begin