diff --git a/src/properties.jl b/src/properties.jl
index 6ad0099..adbe95e 100644
--- a/src/properties.jl
+++ b/src/properties.jl
@@ -11,6 +11,7 @@ struct Arity <: Property
     f
     n::Int
 end
+Base.:(==)(a::Arity, b::Arity) = (a.f, a.n) == (b.f, b.n)
 
 """
     Image <: Property
@@ -22,6 +23,7 @@ struct Image{T} <: Property
     ex::T
     S::AbstractSet
 end
+Base.:(==)(a::Image{T}, b::Image{T}) where {T} = (a.ex, a.S) == (b.ex, b.S)
 isvalid(i::Image{<:Union{Symbolic,Variable,Expr}}, ctx::Context) =
     invoke(isvalid, Tuple{Property,Context}, i, ctx)
 isvalid(i::Image, ctx::Context) =
@@ -39,6 +41,7 @@ struct Signature <: Property
     args::Vector{AbstractSet}
     S::AbstractSet
 end
+Base.:(==)(a::Signature, b::Signature) = (a.f, a.args, a.S) == (b.f, b.args, b.S)
 implies(sig::Signature, a::Arity, ::Context) = a.f == sig.f && a.n == length(sig.args)
 implies(sig::Signature, i::Image{Expr}, ctx::Context) =
     sig.S ⊆ i.S && i.ex.head === :call && length(i.ex.args) == 1 + length(sig.args) &&
@@ -53,9 +56,10 @@ struct Closure <: Property
     f
     S::AbstractSet
 end
+Base.:(==)(a::Closure, b::Closure) = (a.f, a.S) == (b.f, b.S)
 implies(p::Closure, q::Closure, ::Context) = p.f == q.f && q.S ⊆ p.S
 implies(p::Closure, sig::Signature, ::Context) =
-    p.f == sig.f && q.S ⊆ p.S && all(sig.args .⊆ Ref(p.S))
+    p.f == sig.f && sig.S ⊆ p.S && all(sig.args .⊆ Ref(p.S))
 implies(p::Closure, i::Image{Expr}, ctx::Context) =
     p.S ⊆ i.S && i.ex.head === :call && !isempty(i.ex.args) &&
         p.f == i.ex.args[1] && all(isvalid(ctx), Image.(i.ex.args[2:end], Ref(p.S)))
@@ -68,7 +72,8 @@ Representation of function `f` having the associative property, `Associative(f)`
 struct Associative <: Property
     f
 end
-implies(p::Associative, a::Arity, ::Context) = a.n == 2 && a.f == p.f
+Base.:(==)(a::Associative, b::Associative) = a.f == b.f
+implies(p::Associative, a::Arity, ::Context) = a.n ≥ 2 && a.f == p.f
 
 """
     Commutative <: Property
@@ -78,3 +83,4 @@ Representation of function `f` having the commutative property, `Commutative(f)`
 struct Commutative <: Property
     f
 end
+Base.:(==)(a::Commutative, b::Commutative) = a.f == b.f
diff --git a/test/properties.jl b/test/properties.jl
new file mode 100644
index 0000000..6e5e8cd
--- /dev/null
+++ b/test/properties.jl
@@ -0,0 +1,142 @@
+using SpecialSets
+
+
+@testset "Properties" begin
+
+    @testset "Arity" begin
+        @testset "self" begin
+            @test isvalid(Arity(sin, 1), Context(Arity(sin, 1)))
+            @test !isvalid(Arity(sin, 1), Context(Arity(cos, 1)))
+            @test !isvalid(Arity(sin, 1), Context(Arity(sin, 2)))
+        end
+    end
+
+    @testset "Image" begin
+        @vars α
+        @syms a b
+
+        @testset "self" begin
+            @test isvalid(Image(2, Set([2])), Context())
+            @test !isvalid(Image(2, Set([1])), Context())
+            @test isvalid(Image(2, Nonnegative), Context())
+            @test !isvalid(Image(α, Nonnegative), Context())
+
+            @test isvalid(Image(a, TypeSet(Int)), Context(
+                Image(a, TypeSet(Int))
+            ))
+            @test isvalid(Image(a, TypeSet(Number)), Context(
+                Image(a, TypeSet(Int))
+            ))
+            @test !isvalid(Image(a, TypeSet(Int)), Context(
+                Image(a, TypeSet(Number))
+            ))
+            @test !isvalid(Image(b, TypeSet(Int)), Context(
+                Image(a, TypeSet(Number))
+            ))
+        end
+    end
+
+    @testset "Signature" begin
+        @syms a b
+
+        @testset "self" begin
+            @test isvalid(Signature(+, [Positive, Positive], Positive), Context(
+                Signature(+, [Positive, Positive], Positive),
+            ))
+            @test !isvalid(Signature(+, [Positive, Positive], Positive), Context(
+                Signature(*, [Positive, Positive], Positive),
+            ))
+            @test !isvalid(Signature(+, [Positive, Positive, Positive], Positive), Context(
+                Signature(+, [Positive, Positive], Positive),
+            ))
+        end
+
+        @testset "Arity" begin
+            @test isvalid(Arity(+, 2), Context(
+                Signature(+, [Positive, Positive], Positive),
+            ))
+            @test !isvalid(Arity(+, 3), Context(
+                Signature(+, [Positive, Positive], Positive),
+            ))
+        end
+
+        @testset "Image" begin
+            @test isvalid(Image(get(@term 1 + 2), Positive), Context(
+                Signature(+, [Positive, Positive], Positive),
+            ))
+            @test !isvalid(Image(get(@term 1 + 2), Positive), Context(
+                Signature(+, [Positive], Positive),
+            ))
+            @test !isvalid(Image(get(@term 1 + b), Positive), Context(
+                Signature(+, [Positive, Positive], Positive),
+            ))
+            @test isvalid(Image(get(@term 1 + b), Positive), Context(
+                Signature(+, [Positive, Positive], Positive),
+                Image(b, Positive),
+            ))
+        end
+    end
+
+    @testset "Closure" begin
+        @syms a b
+
+        @testset "self" begin
+            @test isvalid(Closure(+, Nonnegative), Context(
+                Closure(+, Nonnegative),
+            ))
+            @test !isvalid(Closure(+, Nonnegative), Context(
+                Closure(*, Nonnegative),
+            ))
+        end
+
+        @testset "Signature" begin
+            @test isvalid(Signature(+, [Positive, Positive], Positive), Context(
+                Closure(+, Positive),
+            ))
+            @test isvalid(Signature(+, [Positive, Positive], Positive), Context(
+                Closure(+, Nonnegative),
+            ))
+            @test isvalid(Signature(+, [Positive, Positive, Positive], Positive), Context(
+                Closure(+, Positive),
+            ))
+        end
+
+        @testset "Image" begin
+            @test isvalid(Image(get(@term 1 + 2), Positive), Context(
+                Closure(+, Positive),
+            ))
+            @test !isvalid(Image(get(@term 1 + 2), Set([3])), Context(
+                Closure(+, Positive),
+            ))
+            @test !isvalid(Image(get(@term 0 + 2), Positive), Context(
+                Closure(+, Positive),
+            ))
+            @test isvalid(Image(get(@term 1 + (2 * 3)), Positive), Context(
+                Closure(+, Positive),
+                Closure(*, Positive),
+            ))
+        end
+    end
+
+    @testset "Associative" begin
+        @testset "self" begin
+            @test isvalid(Associative(+), Context(Associative(+)))
+            @test !isvalid(Associative(+), Context(Associative(*)))
+            @test isvalid(Associative(+), Context(Associative(+), Associative(*)))
+        end
+
+        @testset "Arity" begin
+            @test isvalid(Arity(+, 2), Context(Associative(+)))
+            @test isvalid(Arity(+, 3), Context(Associative(+)))
+            @test !isvalid(Arity(+, 2), Context(Associative(*)))
+        end
+    end
+
+    @testset "Commutative" begin
+        @testset "self" begin
+            @test isvalid(Commutative(+), Context(Commutative(+)))
+            @test !isvalid(Commutative(+), Context(Commutative(*)))
+        end
+    end
+
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index 02a3eac..5777964 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -2,4 +2,5 @@ using Rewrite
 using Test
 
 include("patterns.jl")
+include("properties.jl")
 include("rules.jl")