CHAPTER 1 : Sets
Outline
- Introduction
- Sets and their Represetations
- Set : a set is a well-defined collection of objects.
- Examples of some special sets:
- N : the set of all natural numbers
- Z : the set of all integers
- Q : the set of all rational numbers
- R : the set of real numbers
- Z+ : the set of positive integers
- Q+ : the set of positive rational numbers
- R+ : the set of positive real numbers
- Representation of a set:
- Roster or tabular form is used to list all the elements of a set, separated by commas and enclosed within braces { }.
- Set-builder form is used when all the elements of a set can be defined by a single common property.
- The Empty Set
- Definition 1 : A set which does not contain any element is called the empty set or the null set or the void set.
- Finite and Infinite Sets
- Definition 2 : A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite.
- Equal Sets
- Definition 3 : Two sets A and B are said to be equal if they have exactly the same elements and we write A = B. Otherwise, the sets are said to be unequal and we write A ≠ B.
- Subsets
- Definition 4 :
A set A is said to be a subset of a set B if every element of A is also an element of B.
A ⊂ B if whenever a ∈ A, then a ∈ B.
A ⊂ B if a ∈ A => a ∈ B. - Subsets of set of real numbers
- Intervals as subsets of R
- Power Set
- Definition 5 :
The collection of all subsets of a set A is called the power set of A. It is denoted by P(A).
In P(A), every element is a set.
If A is a set with n(A) = m, then it can be shown that n [ P(A)] = 2m.
- Universal Set
- Venn Diagrams
- Operations on Sets
- Union of sets
- Definition 6 :
The union of two sets A and B is the set C which consists of all those elements which are
either in A or in B (including those which are in both).
A ∪ B = { x : x ∈ A or x ∈ B }
- Properties:
1. A ∪ B = B ∪ A (Commutativity)
2. (A ∪ B) ∪ C = A ∪ (B ∪ C) (Associativity)
3. A ∪ ∅ = A (Law of identity)
4. A ∪ A = A (Idempotent law)
5. U ∪ A = U (Law of U) - Intersection of sets
- Definition 7 :
The intersection of two sets A and B is the set of all those elements which belong to both A
and B.
A ∩ B = { x : x ∈ A and x ∈ B }
- Properties:
1. A ∩ B = B ∩ A (Commutativity)
2. (A ∩ B) ∩ C = A ∩ (B ∩ C) (Associativity)
3. ∅ ∩ A = ∅, U ∩ A = A; (Law of of ∅ and U)
4. A ∩ A = A (Idempotent law)
5. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (Distributive Law: ∩ distributes over ∪) - Difference of sets
A − B = { x : x ∈ A and x ∉ B }
- Complement of a set
- Definition 8 :
Let U be the universal set and A a subset of U. Then the complement of A is the set
of all elements of U which are not the elements of A. Symbolically, A′; is used to
denote the complement of A with respect to U.
A′ = {x : x ∈ U and x ∉ A }.
=> A′ = U − A. - Properties:
1. Complement Laws :
2. De Morgan's Law :
3. Law of double complementation :
4. Laws of empty set and universal set :(i) A ∪ A′ = U
(i) (A ∪ B)′ = A′ ∩ B′
(i) (A′)′ = A
(i) ∅′ = U(ii) A ∩ A′ = ∅
(ii) (A ∩ B)′ = A′ ∪ B′
(ii) U′ = ∅ - Practical Problems on Union and Intersection of Two Sets
