- X → Y (functions)
- f(x)
- f(S)
- f-1(S)
- f-1(Y) = X if X, Y are sets, f : X → Y is a function
- (≺)S→T
- idX
- f∣AB
- f∣AB
- f∣A
- ∘
- (h ∘ g) ∘ f = h ∘ (g ∘ f) if W, X, Y, Z are sets, f : W → X, g : X → Y, h : Y → Z are functions
- idY ∘ f = f if X, Y are sets, f : X → Y is a function
- f ∘ idX = f if X, Y are sets, f : X → Y is a function
- ×
- injective
- surjective
- bijective
- f∣AB is injective if X is a set, A ⊆ X, B is a set, Y ⊆ B, f : X → Y is an injective function
- f∣A is injective if X is a set, A ⊆ X, Y is a set, f : X → Y is an injective function
- f∣AB is surjective ⇒ f is surjective if X is a set, A ⊆ X, B is a set, Y ⊆ B, f : X → Y is a function
- f∣A is surjective ⇒ f is surjective if X is a set, A ⊆ X, Y is a set, f : X → Y is a function
- f∣Af(A) is surjective if X is a set, A ⊆ X, Y is a set, f : X → Y is a function
- g ∘ f is injective if X, Y, Z are sets, f : X → Y is an injective function, g : Y → Z is an injective function
- g ∘ f is surjective if X, Y, Z are sets, f : X → Y is a surjective function, g : Y → Z is a surjective function
- g ∘ f is injective ⇒ f is injective if X, Y, Z are sets, f : X → Y, g : Y → Z are functions
- g ∘ f is surjective ⇒ g is surjective if X, Y, Z are sets, f : X → Y, g : Y → Z are functions
- ↔ (bijections)
- f-1
- f-1(f(x)) = x if X, Y are sets, f : X ↔ Y is a bijection, x ∈ X
- f(f-1(y)) = y if X, Y are sets, f : X ↔ Y is a bijection, y ∈ Y
- (f-1)-1 = f if X, Y are sets, f : X ↔ Y is a bijection
- idX is bijective and (idX)-1 = idX if X is a set
- fn
- fa
- (idX)n = idX if X is a set, n ∈ ℕ
- (idX)a = idX if X is a set, a ∈ ℤ