Relations are subsets of Catesian products (i.e. R ⊆ A x B)
Functions are binary relations and binary relations are sets.
Let f: A --> B be a function. We write f(a) = b when a ∈ A, b ∈ B and (a,b) ∈ f.
Note: A is the domain, B is the codomian, f(A) is the range
To be a function must satisfying the following two:
- for each element in domain, there's a unique element in codomain (pass vertical-line test)
- each element in domain must exist and must appear only once in the binary relation f
Note:
- the preimage may not be unique (when this is the case, the preimage of an element b is the subset S of
the domain A such that f(S)={b})
- There is not necessarily a preimage of an element of the codomain (i.e. f: D --> C, |D| < |C| is ok)
e.g.:
binary relations from A = {1,2,3} to B = {a,b,c}
R1 = {(1,a),(2,a),(3,c)}: this is a function
R2 = {(1,a),(1,b),(2,b),(2,c)}: this is not a function, since 3 does not appear in any pair and 1 and 2 both
appear twice each
R3 = {(1,c),(2,a)}: this is not a function, since 3 does not appear in any pair
y = f(x) (x ∈ A) ⇒ y ∈ f(A)
y ∈ f(A) ⇒ y = f(x) (x ∈ A); likewise, y ∈ f(A ∩ B) ⇒ y = f(x) (x ∈ A and x ∈ B)
