Lecture # 1
Elementary set theory
In this lecture, we shall discuss elementary concepts of set theory
Set
A set is any well-defined list, collection, or class of objects.
Examples: 𝐴 = {2, 4, 6, … }, 𝑁 = {1, 2, 3, … }.
Notations
Sets will usually denoted by capital letters such as A, B, X,….
The elements in sets are usually denoted by lower case letters such as a, b, x,
y,….
Representation of set:
1. Tabular form 2. Set-builder form
Finite and Infinite sets
A set is finite if it consists of a specific number of different elements, i.e., if in
counting the different members of the set the counting process can come to an end.
Otherwise a set is infinite.
Equality of sets:
Set 𝐴 is equal to set 𝐵 if they both have the same members.
Subsets:
If every element in a set 𝐴 is also a member of set 𝐵 then 𝐴 is called a subset of 𝐵.
Theorem:
(i)
(ii)
Null set is subset of every set.
If 𝐴 is a subset of 𝐵 and 𝐵 is a subset of 𝐶 then 𝐴 is a subset of 𝐶.
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Proof:
(i)
(ii)
Let A is any set. To prove ∅ ⊆ 𝐴. Let𝑥 ∉ 𝐴. Then 𝑥 ∉ ∅. This
implies ∅ ⊆ 𝐴.
Let 𝑥 ∈ 𝐴. As 𝐴 ⊆ 𝐵, so 𝑥 ∈ 𝐵 and 𝐵 ⊆ 𝐶 ⇒ 𝑥 ∈ 𝐶.
Set of sets:
If element in a set are also sets then set is called set of sets.
Power set
The family of all subsets of a set S is called power set of S.
Venn-Euler Diagrams
A simple and instructive way of illustrating the relationships between sets is in
the use of so-called Venn-Euler diagrams. Here we represent a set by a plane
area,
Line Diagrams
Another useful and instructive way of illustrating the relationships between sets
is in the use of so-called line diagrams. If A is a subset of B then we write B on
a higher level.
Basic set operations:
(i)
Union:
The union of two sets A and B is the sets of all elements which belong to
A or to B or to both.
𝐴 ∪ 𝐵 = {𝑥: 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵}.
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Properties:
(a) 𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴
(b) 𝐴 ⊆ 𝐴 ∪ 𝐵 𝑎𝑛𝑑 𝐵 ⊆ 𝐴 ∪ 𝐵
(c) 𝐴 ⊆ 𝐵 ⇒ 𝐴 ∪ 𝐵 = 𝐵
(d) 𝐴 = 𝐴 ∪ 𝐴
(ii) Intersection:
The intersection of two sets A and B is the sets of all elements which
belong to A and B.
𝐴 ∩ 𝐵 = {𝑥: 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵}.
Properties:
(a) 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴
(b) 𝐴 ⊇ 𝐴 ∩ 𝐵 𝑎𝑛𝑑 𝐵 ⊇ 𝐴 ∩ 𝐵
(c) 𝐴 ⊆ 𝐵 ⇒ 𝐴 ∩ 𝐵 = 𝐴
(d) 𝐴 = 𝐴 ∩ 𝐴
(iii)
Difference:
The difference of two sets A and B is the set of elements which belong to
A but which do not belong to B.
𝐴 − 𝐵 = {𝑥: 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵}.
Properties:
(a) 𝐴 − 𝐵 ⊆ 𝐴
(b) (𝐴 − 𝐵) ∩ 𝐵 = ∅ .
(iv) Complement:
The complement of a set A is the set of those elements of universal set U
which do not belong to A.
𝐴′ = {𝑥: 𝑥 ∈ 𝑈 𝑎𝑛𝑑 𝑥 ∉ 𝐴}.
Properties:
(a) 𝐴 ⊆ 𝐵 ⇒ 𝐵′ ⊆ 𝐴′
(b) 𝐴 − 𝐵 = 𝐴 ∩ 𝐵′
(c) 𝐴 ∪ 𝐴′ = 𝑈 𝑎𝑛𝑑 𝐴 ∩ 𝐴′ = ∅
(d) (𝐴 ∪ 𝐵)′ = 𝐴′ ∩ 𝐵′
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Exercise:
Prove all properties of Union, Intersection, difference and complement operations.
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