Scribd
Upload
2 views2 pages

Math113 6

The document introduces fundamental set notation and concepts, including subsets and proper subsets, as well as functions and their properties such as injectivity, surjectivity, and bijectivity. It explains how functions map between sets and discusses the concept of cardinality, stating that two sets have the same cardinality if there exists a bijection between them. An example is provided to illustrate the relationship between the sets of integers and positive integers.

Uploaded by

ksh.singh011
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as TXT, PDF, TXT or read online on Scribd
2 views2 pages

Math113 6

The document introduces fundamental set notation and concepts, including subsets and proper subsets, as well as functions and their properties such as injectivity, surjectivity, and bijectivity. It explains how functions map between sets and discusses the concept of cardinality, stating that two sets have the same cardinality if there exists a bijection between them. An example is provided to illustrate the relationship between the sets of integers and positive integers.

Uploaded by

ksh.singh011
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as TXT, PDF, TXT or read online on Scribd
You are on page 1/ 2

Lecture 1: January 23 2

Some more general notation includes

a ∈ S a is an element of S
a 3 S a is not an element of S
∅ the empty set (the set with no elements)

A ⊆ B A is a subset of B
A ⊂ B A is a proper subset of B

If A is a subset of B, that means every element of A is in B: ∀x ∈ A, x ∈ B. If A


is a proper subset of B,
then A ⊆ B and A 6= B.

As an example, S = {0, 1} has the subsets ∅, 0, 1, 0, 1, so there are 4 subsets and


3 proper subsets.

1.3.2 Functions

A function f : X → Y, x→ f(x) goes between sets X and Y , x ∈ X and f(x) ∈ Y . If A


⊆ X, then

f(A) = {f(x) | x ∈ A}

and f(X) is referred to as the image of f .

If B ⊆ Y and f−1(B) = {x ∈ X | f(x) ∈ B}, then f−1(Y ) is the inverse image or


preimage of f .

Some important properties of functions are as follows:

1. f is injective or one-to-one if f(x1) = f(x2) =⇒ x1 = x2. Different elements of


X map to different
elements of Y . For example, f : R→ R, x→ x2 is not injective, because f(3) = f(−3)
= 9. However,
the function f : R+ → R+, x→ x2 is injective.

2. f is surjective or onto if f(X) = Y , that is, every value in Y is attained by f


. For example, f :
R → R, x → x2 is not surjective, because @x ∈ R such that f(x) = −1. However, the
function
f : R+ → R+, x→ x2 is surjective.

3. f is bijective if it is injective and surjective. If f is bijective, there


exists an inverse correspondence
f−1Y → X, mapping y → the unique x such that f(x) = y.

We have seen that f : R+ → R+, x → x2 is injective and surjective, which suggests


that we can make this
inverse relationship: f−1 : R+ → R+, y → √y.

1.3.3 Cardinality

We say that X and Y have the same cardinality if there exists a bijection between
them. If the sets are
finite, they have the same number of elements.
We claim that Z and Z+ have the same cardinality. To prove or disprove this, we
want to build a bijection
Z→ Z+. This bijection can be constructed as

You might also like