1- Definition of Cardinality
The cardinality of a set refers to the total number of distinct elements in that set. It is
denoted by ∣A∣ or n(A), where A represents the set in question. For instance,
if A={1,2,3,4,5}, then the cardinality of set A is ∣A∣=5 because it contains five elements
Types of Cardinality
Finite Cardinality: A set is said to be finite if it has a limited number of
elements. For example, the set B={a,b,c} has a cardinality of 3, as it contains
three elements
Infinite Cardinality: A set is infinite if it has an unbounded number of
elements. Infinite sets can be further classified into:
Countably Infinite: A set is countably infinite if there is a one-to-one
correspondence (bijection) between its elements and the natural numbers N.
Examples include the set of integers Z and the set of rational numbers Q, both
of which have cardinality ℵ0​ (aleph null)
Uncountably Infinite: A set is uncountable if it is not possible to establish a one-to-one
correspondence with the natural numbers. An example of an uncountable set is the set
of real numbers RR, which has a larger cardinality than NN
Examples of Cardinality
•Countable Sets: The set of natural numbers N={1,2,3,…}N={1,2,3,…} is countable
with cardinality ∣N∣=ℵ0∣N∣=ℵ0​.
•Uncountable Sets: The interval [0,1][0,1] is uncountably infinite, meaning it has a
greater cardinality than any countable set
Comparing Cardinalities
Two sets AA and BB can be compared using cardinalities:
If ∣A∣=∣B∣∣A∣=∣B∣, there exists a bijection between the two sets.
•If ∣A∣>∣B∣∣A∣>∣B∣, set AA cannot be fully mapped into set BB
•If ∣A∣<∣B∣∣A∣<∣B∣, there is an injection (one-to-one mapping) from AA to BB.
Formal Definition
Mathematically, a function f:A→B is called injective (or one-to-one) if for
any x1,x2∈Ax1​,x2​∈A:
•If f(x1)=f(x2)f(x1​)=f(x2​), then x1​=x2​.
This can also be expressed equivalently by saying
If ( x_1
eq x_2 ), then ( f(x_1)
eq f(x_2) )
Properties of Injective Functions
Distinct Outputs: Each element in the codomain (the set of possible output values) is the
image of at most one element from the domain (the set of input values). This implies that no
two different inputs will map to the same output
Horizontal Line Test: For functions defined on the real numbers, a graphical way to
test injectivity is the horizontal line test. If any horizontal line intersects the graph of the
function more than once, then the function is not injective
Examples:
•The function f(x)=x+1 is injective because different inputs will yield different outputs.
Conversely, g(x)=𝑥 2 is not injective when defined over all real numbers because
g(2)=g(−2)=4g(2)=g(−2)=4
Definition
A function f:A→B is called surjective (or onto) if for every element y in the codomain B, there exists at least
one element x in the domain A such that f(x)=y. In simpler terms, every possible output in B is mapped from
some input in A
Properties of Surjective Functions
Equal Range and Codomain: The image (or range) of the function is exactly equal to the
codomain. Thus, every element of the codomain has a preimage in the domain
Multiple Preimages: A surjective function can map multiple inputs to the same output. This means
that two distinct inputs in the domain can yield the same value in the codomain
Right Inverses: Every surjective function has a right inverse, meaning there exists a function that
can "reverse" the mapping, at least partially
Examples
Linear Functions: The function f(x)=2x+1f(x)=2x+1 from real numbers to real numbers is surjective,
as for any y, we can find x=y−12x=2y−1​ such that f(x)=y
Quadratic Functions: The function g(x)=𝑥 2 is not surjective when viewed from real numbers to real
numbers since negative numbers cannot be outputs. However, when viewed from real numbers to nonnegative real numbers, it is surjective
Definition of Bijection
A bijective function, or bijection, establishes a one-to-one correspondence between two sets A and B. This
means that for every element in set A, there is a unique corresponding element in set B, and vice versa. In
mathematical notation, a function f:A→B is bijective if:
•Injective (One-to-One): No two different elements in A map to the same element in B. Formally,
if f(a1)=f(a2)f(a1​)=f(a2​), then a1=a2
Surjective (Onto): Every element in B is the image of at least one element in A. This means the range
of ff covers every element in B
Examples of Bijection
•Identity Function: The function f(x)=x is a bijection between any set and itself.
Linear Functions: For f(x)=2x+1 every input x has a unique output, and every real number can be achieved
as an output; thus, it is bijective as both injective and surjective