Course Material 1.8 Countable sets and Uncountable sets.
Injective functions
Let 𝑓: 𝐴 → 𝐵 be a function. We say that 𝑓 is a one –one function or is an injective
function if for any two points 𝑥1 and 𝑥2 in 𝐴, with 𝑥1 ≠ 𝑥2 , then 𝑓 𝑥1 ≠ 𝑓(𝑥2 ).
(Equivalently, if 𝑓 𝑥1 = 𝑓 𝑥2 then 𝑥1 = 𝑥2 ). This means that distinct elements of the
domain have distinct images in the co domain.
Surjective functions
A function 𝑓: 𝐴 → 𝐵 is said to be surjective or we say that 𝑓 maps 𝐴 onto 𝐵 if every
element 𝑦 of the co domain 𝐵 is the image of some element of 𝐴. This means that if 𝑦 ∈ 𝐵
then there exists at least one 𝑥 ∈ 𝐴. Such that 𝑓 𝑥 = 𝑦
Bijective functions.
A function 𝑓: 𝐴 → 𝐵 is said to be a surjective function if it is both injective and surjective. (
i.e, both one – one and onto). Bijective functions are also called one – one correspondences.
Examples
1. Let 𝑓: 𝐴 → 𝐵, 𝐴 = [0,1] and 𝐵 = [1, 3] be defined by 𝑓 𝑥 = 2𝑥 + 1
If 𝑓 𝑥1 = 𝑓 𝑥2 then 2𝑥1 + 1 = 2𝑥2 + 1 ⇒ 2𝑥1 = 2𝑥2 ⇒ 𝑥1 = 𝑥2 .
Hence 𝑓 is one - one
𝑦−1
Let 𝑦 ∈ 𝐵 and suppose for some 𝑓 𝑥 = 𝑦. Then 2𝑥 + 1 = 𝑦 and Hence 𝑥 = 2
𝑦−1
Also if 𝑦 ∈ 𝐵, we have 1 ≤ 𝑦 ≤ 3 ⇒ 0 ≤ 𝑦 − 1 ≤ 2 ⇒ 0 ≤ ≤ 1 or 0 ≤ 𝑥 ≤ 1.
2
𝑦−1
This shows that for each 𝑦 ∈ 𝐵, 𝑥 = is an element of 𝐴 such that 𝑓 𝑥 = 𝑦 and
2
hence proving that 𝑓 is surjective.
2. The function 𝑓: ℝ → ℝ defined by 𝑓 𝑥 = 𝑥 2 is neither injective nor is surjective.
Remark:
Suppose 𝐴 and 𝐵 are finite sets. Let us use the symbol 𝐴 and 𝐵 for the number of
elements in 𝐴 and the number of elements in 𝐵
Injective functions
Let 𝑓: 𝐴 → 𝐵 be a function. We say that 𝑓 is a one –one function or is an injective
function if for any two points 𝑥1 and 𝑥2 in 𝐴, with 𝑥1 ≠ 𝑥2 , then 𝑓 𝑥1 ≠ 𝑓(𝑥2 ).
(Equivalently, if 𝑓 𝑥1 = 𝑓 𝑥2 then 𝑥1 = 𝑥2 ). This means that distinct elements of the
domain have distinct images in the co domain.
Surjective functions
A function 𝑓: 𝐴 → 𝐵 is said to be surjective or we say that 𝑓 maps 𝐴 onto 𝐵 if every
element 𝑦 of the co domain 𝐵 is the image of some element of 𝐴. This means that if 𝑦 ∈ 𝐵
then there exists at least one 𝑥 ∈ 𝐴. Such that 𝑓 𝑥 = 𝑦
Bijective functions.
A function 𝑓: 𝐴 → 𝐵 is said to be a surjective function if it is both injective and surjective. (
i.e, both one – one and onto). Bijective functions are also called one – one correspondences.
Examples
1. Let 𝑓: 𝐴 → 𝐵, 𝐴 = [0,1] and 𝐵 = [1, 3] be defined by 𝑓 𝑥 = 2𝑥 + 1
If 𝑓 𝑥1 = 𝑓 𝑥2 then 2𝑥1 + 1 = 2𝑥2 + 1 ⇒ 2𝑥1 = 2𝑥2 ⇒ 𝑥1 = 𝑥2 .
Hence 𝑓 is one - one
𝑦−1
Let 𝑦 ∈ 𝐵 and suppose for some 𝑓 𝑥 = 𝑦. Then 2𝑥 + 1 = 𝑦 and Hence 𝑥 = 2
𝑦−1
Also if 𝑦 ∈ 𝐵, we have 1 ≤ 𝑦 ≤ 3 ⇒ 0 ≤ 𝑦 − 1 ≤ 2 ⇒ 0 ≤ ≤ 1 or 0 ≤ 𝑥 ≤ 1.
2
𝑦−1
This shows that for each 𝑦 ∈ 𝐵, 𝑥 = is an element of 𝐴 such that 𝑓 𝑥 = 𝑦 and
2
hence proving that 𝑓 is surjective.
2. The function 𝑓: ℝ → ℝ defined by 𝑓 𝑥 = 𝑥 2 is neither injective nor is surjective.
Remark:
Suppose 𝐴 and 𝐵 are finite sets. Let us use the symbol 𝐴 and 𝐵 for the number of
elements in 𝐴 and the number of elements in 𝐵
