Vocabulary Flashcards
Statement
A statement is a declarative sentence that is either true or false.
Set
a set is a collection of elements.
Subset
A set B is a subset of a set C if every element of B is in C
Union
B ∪ C = {x | x ∈ B or x ∈ C}
Intersection
B ∩ C = {x | x ∈ B and x ∈ C}
Cartesian Product
The Cartesian product of sets B and C is the set
B × C = {(x, y) | x ∈ B and y ∈ C}.
The elements of B × C are ordered pairs of elements.
Function
A function f from a set B to a set C, denoted f : B → C, is a rule that assigns each element b in B to exactly one element c of C.
Note:
B is called the domain and C is the range of f .
c = f (b) is called the value of f at b or the image of b under f .
The function f is sometimes called a map or mapping.
Composition of Functions
Given functions f : B → C and g : C → D, let h : B → D be given by h(x) = g(f (x)) for all x ∈ B. The composite function h is also denoted by g ◦ f . Thus (g ◦ f )(x) = g(f (x)).
Injective
A function f : B → C is injective (or a one-to-one mapping) if f maps distinct elements of B to distinct elements of C.
Surjective
A function f : B → C is surjective (or an onto mapping) if every element of C is the image under f of at least one element of B.
Image of function
If f : B → C is a function, then the image of f is the subset of C:
Im f = {c ∈ C | c = f (b) for some b ∈ B} = {f (b) | b ∈ B}.
Image of subset
If f : B → C is a function and S is a subset of B, then the image of S under f is the
subset of C:
f (S) = {c ∈ C | c = f (b) for some b ∈ S} = {f (b) | b ∈ S}.
Bijective
A function f : B → C is bijective (or a bijection) if it is both injective and surjective.
Inverse
If f : B → C is a bijection, then the map g in the Theorem above is called the inverse function of f and is sometimes denoted by f −1.
Sets of equal cardinality
Two sets B and C have the same cardinality if there is a bijection f : B → C. We write in this case |B| = |C|.
