Set

Introduction to Sets

Set is a collection of elements with the common properties.

Let’s take the set as stationeries.
The elements would be pen, correction liquid, ruler, eraser, pencils and many more.
It’s just things grouped together with a certain property in common (in this case all of them are stationeries.)

Way of listing the elements of Sets

- The Roster Method

By listing all the element members of a set, and announcing the common properties of the member elements as the name of the Set. Such that:

The set O of even positive integers less than 10 can be expressed by O = {2,4,6,8}
The set of positive integers less than 100 can be denoted by {1, 2, 3, . . . , 99}

- The Set Builder Notation

Properties of the member elements must be made known in the expression and elements of the set must be characterized.

For example, the set O of all even positive integers less than 10 can be written as

O = {x | x is an even positive integer less than 10}

Or, specifying the universe as the set of positive integers, as

O = {x ∈ Z+ | x is even and x < 10}

The set Q+ of all positive rational numbers can be written as

Q+ = {x∈ R | x = p/q, for some positive integers p and q}

The example above can be defined as

 Stationery = {pen, correction liquid, ruler...}

Ellipsis is “…” that symbolizes “and the list goes on”.

If used in the middle, such as {a, b, c … x, y, z} It’s used to save writing long list.

There can also be sets that do not have common properties. And they are just given like that.

For example, 

A = {13, 23, 31, 42}

B = {1, 2, 3, 4, 5, 6, 7, 8}

Set A does not have any interconnection among its elements, while Set B is just composed of numbers according to increasing order.

Finite / Infinite Set

Prime Numbers = {2, 3, 5, 7…}

Fingers ={Thumb, Index, Middle, Ring, Pinky}

The set for prime numbers are infinite set, which have no limitation of length it is going.
While the set for the name of fingers is limited to only 5, therefore a finite set.

We use ∈ to show that an element is a subset of a set, and ∉to show that an element is not available inside a specific set.

Set A is {13, 23, 31, 42}. You can see that 13 ∉ A, but 5 ∈ A

Empty Set / Null Set

It’s a set without any elements in it. It can be represented as {} or Ø.

And, although weird, an empty set is a subset of every set, including the empty set itself.

Set of numbers

Natural numbers is the term we used to refer positive integers, excluded 0.

N  = { 1, 2, 3, 4, 5, …. }

Whole numbers similar to natural numbers, but 0 included.

W  = { 0, 1, 2, 3, 4, 5, …. }

Integers are a set of numbers with consist of positive and negative numbers.

Z = {  …, -2, -1, 0, 1, 2, … }

Positive integers are a set of integers that consists of positive values.

Z⁺ = {  1, 2, 3, 4, … }

Negative integers are a set of integers that consist of negative values.

Z^- = { … , -4, -3, -2, -1 }

Rational numbers are numbers with a fixed pattern of decimals.

Example :  0.33333

Irrational Numbers which have no fixed pattern of decimal.

example: π=1.342

All the sets of numbers above are classified as real numbers. Real numbers are all numbers except for imaginary numbers.

Set Equality

Two sets are equal to each other if and only if they have the same members of element in their set, regardless of the sequence and order.

A is the set whose members are the first four positive whole numbers

B = {4, 2, 1, and 3}

Both set pointing to the number 1, 2, 3 and 4. So yes, they are equal sets.

Venn Diagram

Sets can be represented graphically using Venn diagrams (Named after the English mathematician John Venn,1881). In Venn diagrams the universal set U, which contains all the objects under consideration, is represented by a rectangle. Inside this rectangle, circles or other geometrical figures are used to represent sets. Sometimes points are used to represent the particular elements of the set. Venn diagrams are often used to indicate the relationships between sets.

Examples of Venn Diagram:

                                                

Subset

B is a subset of A if and only if all the elements in set B are present in set A.

A is a set of all even numbers.

B = {2, 4, 6}

2 are inside both sets. 4 are also inside both set. So does 6.

So, we can conclude that B is a subset of A.

Example:

Universal Set

It is a set that contain everything (that is relevant to the particular problem) and is denoted as U.

Cardinality 

The cardinality is the number of elements in any given set(s).  |A| is used to represent the numbers contained in a particular set, in this case A.

Examples :

|{3, 4, 5, 6}| = 4

This is because the set contains 4 elements inside.

Power Set

The power set of S, is all the subset of set S, including the empty set and set S itself.

If set S has a cardinality of |S| = n, then the number of subset of |P(S)| = 2ⁿ.
So, if a set has a cardinality of 0, then 2 ⁰ = 1. It means that the power set of an empty set is the empty set itself.

When the cardinality of the set is, let’s say 2, then it will be 2² =4, as |2ⁿ|=|2 ²|.

The power set of a set S, together with the operations of union, intersection and complement can be viewed as the prototypical example of a Boolean algebra.

Set Operation

1. UNION

Let A and B be sets. The union of the sets A and B @ A ∪ B, is the set that contains

those elements that are either in A or in B, or in both.

In mathematical definition, A ∪ B = { x | x ∈ A ∨ x ∈ B }

The Venn diagram below represents the union of two sets A and B. The area that represents A ∪ B is the shaded area within either the circle representing A or the circle representing B.

Example :

The union of the sets {w, e, b, u, n, t, u} and {b, e,s, t} is the set {w, e, b, u, n, t, u, b, e, s, t}

@

{1, 3, 5} ∪ {1, 2, 3} = {1, 2, 3, 5}.

2.  INTERSECTION

The intersection of sets A and B @ A ∩ B, is the set containing

those elements in both sets A and B.

In mathematical definition, A ∩ B = { x | x ∈ A ∧ x ∈ B }

Example :

The intersection of the sets {m, a, t, h } and {e, x, c, e, l, l, e, n, t} is the set {t}

@

{1, 3, 5} ∩ {1, 2, 3} = {1, 3}.

3. DISJOINT

Two sets are called disjoint if their intersection is empty set.

Example :

Let A = {1, 3, 5, 7, 9} and B = {2, 4, 6, 8, 10}

Since A ∩ B = ф

Then A and B are disjoint.

4. SET DIFFERENCE

·     Let A and B be sets. The difference of A and B, denoted by A minus B   (A-B), is the set containing those elements that are in A but not in B.

·     The difference of A and B is also called the complement of B with respect to A and sometimes denoted by A/B.

·     An element X belongs to the difference of A and B if and only if X ∈ A and X ∉ B . This tells us   that A – B = { X | X  ∈ A ᴧ X  ∉ B}

·     Example :       A = {1 , 3 , 5}           B = {1 ,2 , 3}

A – B = {1 , 3 , 5} – {1 , 2 , 3}  = {5}

B – A = {1 , 2 , 3} – {1 , 3 , 5} = {2}

                                               

                                                                 A-B is shaded

          Venn Diagram for Difference of A and B

5. SET COMPLEMENTARY

Let U be the universal set. The complement of the set A, denoted by A' , is the complement of A with respect to U. Therefore, the complement of the set A is U - A.

·        An element belongs to A'  if X ∉ A. This tells us that A' = { X  ∈ U | X ∉ A}

·        Example :     U= {1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10}

A= {1 , 3 , 5 , 7 , 9}

A'= {2 , 4 , 6 , 8 , 10}

 

                    A' is shaded

Venn Diagram for the Complement of the set A

6. CHARACTERISTICS OF SET

Identity laws

A∩U = A

A∪ф = A

Domination laws

A∪ U =U

A∩ ф = ф

Idempotent laws

A∪A=A

A∩A=A

Complementation law

(A’)’= A

Commutative laws

A∪ B = B∪A

A∩B = B∩ A

Associative laws

A∪(B∪C) = (A∪B)∪C

A∩(B∩C) = (A∩B)∪C

Distributive laws

A∪(B∩C) = (A∪B)∩(A∪C)

A∩(B∪C) = (A∩B)∪(A∩C)

De Morgan’s laws

(A∩ B)’ = A’∪B’

(A∪B)’ = A’∩B’

Absorption laws

A∪ (A∩B) = A

A∩(A∪B) = A

Complement laws

A∪A’ = U

A∩A’ = ф

GENERALIZED UNIONS & INTERSECTION.

Union’s definition and notation:

The union of a collection of sets is the set that contains those elements that are members of

at least one set in the collection. The notation:

A₁ U A₂ U….U A_n=

Intersection’s definition and notation:

The intersection of a collection of sets is the set that contains those elements that are members of all the sets in the collection. The notation:

A₁ ∩ A₂ ∩….∩ A_n=

CARTESIAN PRODUCT

 Consider A & B is a set. Cartesian product of A & B, A×B is set of ordered pair (a,b), where aϵA and bϵB. Hence,

A×B= {(a,b) | a ϵ A ˄ b ϵ B}.

§  E.g:

A={1,2} , B={4,5}

A×B={(1,4),(1,5),(2,4),(2,5)}

Cartesian product of set A₁, A₂,….,A_n.

A₁ × A₂ × …. × A_n={(a₁, a₂,…. a_n)|a_i ϵ A_i}