SET ANALYSIS:DEFINITION AND TYPES OF SET

Definition of set

A set is nothing but a well defined and well-distinguished collection of objects and it contains number, words, picture, etc.

What do we mean by a well-defined collection of objects?  If we define a particular set suppose set of planets in our solar system we don’t have any doubt about which planets are there in our solar system.  But if we search for a good football player which player will belong to that set, it’s a big question. We may find somebody a good player but in view of another person, he may not be a good player. So the set of planets in the solar system is a very defined set we all know there are only eight planets in our solar system. So anything other than those eight does not belong to that set.  But if we talk about a good player in football, the set of good players in football may value from person to person. We may like Messi but there can be another person who likes Ronaldo. So the set of good players in football or good players in cricket or our favorite flower or the good book in the library all these sets which vary from a person to person cannot be called sets.  So a set is a well-defined collection of objects which is true for every person or every situation.  So a set is a very well defined collection of objects.

 Well defined collection of objects/Set Not well-defined collection of objects/Vary person to person/Not a set
1.The set of planets of our solar system.                                                                                                            2.The colors of the rainbow

1.A favorite food of children

2.The  Favourite flavors of ice-cream

 

Methods of representing a set

There are actually three methods by which we can represent a set.

a)Descriptive method:

The descriptive method actually defines the set in proper words.

Example-1: The set of all the colors of the rainbow,

Example-2: The set of natural numbers less than 20.

b) Rooster method:

In this method, we manually list all the members or all the elements of the set.

Example-1:  a set a which actually lists all the alphabets of a flower we can write it as A={f,l,o,w,e,r}.

Example-2: a set of all the alphabets of the word floor which we can write as B={f,l,o,r}

Though the word floor consists of the alphabet f, l,double-o, and r, we can not write double o because any element can appear only once in a set.  So this is the rooster method and it is a very crude method in which we have to write all the elements manually.

C)Set builder method/Rule method:

In this method, we actually write a rule which the whole set follows.  Actually this is a very smart method.

Example-1: Suppose we want to write all the natural numbers which are less than 30. We can write by this method S={x/x≤30 and x ε N} which means, S has an element x  where x is less than equal to 30, and X belongs to or X is an element of set S of natural numbers. The set of natural numbers is represented as N.

example-2.  If we want to make or define a set of even numbers which are less than 50. We can write this as E={x/x≤50 and x is divisible by 2 and x ε N}. This is the set builder method or rule method of writing a set.

 

TYPES OF SET:

Finite set:

 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.  That means the set must be `countable’ and it has limits’.

Example-1: E={x/x≤50 and x is divisible by 2 and x is Natural number}.
Example-2: C={V,B,I,G,Y,O,R}, where C is the set of all the colors of the rainbow.

 

Infinite set:

A set that has infinite numbers of elements is called an infinite set.  That is the set which is related to `uncountable’ and `limitless’ is an infinite set.

Example-1: A={all the natural numbers} and We can write it as

A={0,1,2,3—∞}

Example-2: The set of all integers, That means B={…, -1, 0, 1, 2, —-}

Note:

All infinite sets cannot be expressed in roster form.

For example, The set of real numbers. Since the elements of this set do not follow any particular pattern.

Null Set:

A set with no elements is called a Null set.

This is also known as the Empty Set or void set. There is not any elements in it. 

It is represented by which  is read as phi Or by {} (a set with no elements)

Example-1: A={The people of 200 years old}.There are no people aged 200 years old.
Example-2: B={ The set of piano keys on a guitar}. Actually  “There are no piano keys on a guitar!”
Note:

∅ ≠ {0} ∴ has no element.

{0} is a set that has one element 0.

Unique set/Singleton Set:

A set which contains only one element is called a singleton set.

For example:

  1. A = {x : x is neither prime nor composite}It is a singleton set containing one element, i.e., 1.2.B = {x : x is a whole number, x < 1}This set contains only one element 0 and is a singleton set.3. Let B = {x : x is a even prime number}Here B is a singleton set because there is only one prime number which is even, i.e., 2.

Equal set/Equality/Identical set

If two sets  have precisely the same elements and same cardinal number we call them equal set.

Two sets E and F are equal if and only if every element in E belongs to F and every element in F belongs to E.

Example-1:  E and F are equal where:
  • E= {1,2,3}
  • F = {2, 1, 3}

Here E and F have the same elements 1,2 and 3 and the cardinal number of set E  is n(E)=3 and the cardinal number of set F is n(F)=3 are equal.

So Set E and F are equal and the equals sign (=) is used to show equality. We can write it as E=F.

Example-2 : Let G={3,9,11}    and  H={11,3,9};

Here G=F, as they have the same elements and same cardinal number order, need not be the same.

Example-3: Are these sets equal?
  • A is {1², 2², 3²}
  • B is {1, 4,9}

Yes, they are equal!

They both contain exactly the members 1, 4 and 9

 

Equivalence /Equivalent Sets:

Two sets A and B are said to be equivalent if their cardinal number is same, i.e., n(A) = n(B) where the elements may or may not have the same. The symbol for denoting an equivalent set is ‘↔’.

 Example-1

A = {1, 2, 3} Here n(A) = 3

B = {p, q, r} Here n(B) = 3
Here the cardinal number of A is equal to the cardinal number of B.But the elements are not the same of the two sets.
Therefore, we can say set A is equal to set B i.e. A ↔ B.

Example-2: Let A={1,2,3} , B={3,4,5,9} , C={3,9,2,8} , and    D={2,3,4}

Here, n(A)=3       n(B)=4          n(C)=4             n(D)=3

 and here,     A is equivalent to D     and    B is equivalent to C

Note:

1. Equal sets are always equivalent

2. Equivalent sets may or may not be Equal.

Subset

Suppose the boy has had three options to choose from chocolate, butterscotch, and Vanilla.

Let’s just refer to them as c, b, and v. If he can have only one kind of ice cream at a time it means he can have either chocolate or butterscotch or vanilla. Then the set will be with any one flavor. But what if he feel like having two at a time?  The different possibilities he can have chocolate and butterscotch together or he may want chocolate and vanilla together or any other combination possible. Then the set will be with two elements and if he is unwell and the doctor has advised him against having any ice cream, in that case what we have is an empty set and the original set is with  chocolate, butterscotch, and vanilla. i.e.{c,b,v}. This is the set of all kinds of ice cream. Such this way the subsets of entire set C= {c,b,v} will be the sets below:

Anyone      Any two              Three                                 None

{c}               {c,b}                    {c,b,v}                                 {   }

{b}               {c,v}                 Entire set                           Empty set

{v}             {b,v}                (It is subset of itself)     (It is also subset of entire set)

Each set above  are a part of the main set C and in mathematics, we say that

each set is a subset of the main set.i.e. the subsets of  the set C are

{c}  ,{b}, {v} , {c,b}   ,{c,v}, {b,v}    {c,b,v} ,  {   }

 If E and F are two sets, and every element of set E is also an element of set F, then E is called a subset of F and we write it as E F or F E

The symbol  stands for ‘ subset of’ or ‘is contained in’.

For example; Let E = {2, 4, 6},F = {6, 4, 8, 2}.Here E is a subset of F. Since, all the elements of set E are contained in set F.But F is not the subset of E. Since, all the elements of set F are not contained in set E.

Note:

• Every set is a subset of itself, i.e., E ⊂ E, F ⊂ F.

• Null set is a subset of every set.

• E ⊆ F means E is a subset of F or E is contained in F.

• F ⊆ E means E contains F.

. The set N of natural numbers is a subset of the set Z of integers and we can write it as N ⊂ Z.

. Let E = {2, 4, 6} ;F = {x / x is an even natural number less than 8}

Here E ⊂ F and F ⊂ E.Hence, we can say E = F .

 Let E = {1, 2, 3, 4} , F = {4, 5, 6, 7}. Here E ⊄ F
[ denotes ‘not a subset of’]
Super Set:

Whenever a set E is a subset of set F, we say the F is a superset of E and we write, F ⊇ E.

Symbol ⊇ means  ‘is a superset of’

For example;

Let, E = {a, e, i, o, u}, F = {a, b, c, …………., z}

Here E ⊆ F i.e., E is a subset of F but F ⊇ E i.e., F is a superset of E

`Proper Subset:

If E and F are two sets, then E is called the proper subset of F if E ⊆ F but F ⊇ E i.e., E ≠ F. The symbol ‘⊂’ is used to denote proper subset. Symbolically, we write E ⊂ F.

For example;

  1. Let  E = {1, 2, 3, 4}. Here n(E) = 4; F = {1, 2, 3, 4, 5}; Here n(F) = 5.We observe that all the elements of E are present in F but the element ‘5’ of F is not present in E.So, we say that E is a proper subset of F.
    Symbolically, we write it as E ⊂ F
    2 .E = {p, q, r}, F = {p, q, r, s, t}

    Here E is a proper subset of F as all the elements of set E are in set F and also E ≠ F.

Notes:

1.No set is a proper subset of itself.

2. The empty set is a proper subset of every set.

 

Comparability

Two sets A and B are said to be comparable if A ⊂ B or B ⊂ A.

That is if one of the sets is a subset of the other set we call them comparable set. Moreover, two sets A and B are said to be not comparable if A ⊄ B and also B ⊄ A

Example:

1.A={a,b} and B={a,b,c}.Then A is comparable to B,since A is a subset of B.

2.R={a,b} and S={b,c,d}.Then R and S are not comparable, since a ε R and a∉ S and cε S, and c∉ R.

Disjoint set:

Two sets A and B are said to be disjoint sets if they have no element in common,i.e. if no element of A is in B and no element of B is in A, then we say that A and B are Disjoint set. Equivalently, disjoint sets are sets whose intersection is the empty set.

Example-1:A= {1, 2, 3} and B= {4, 5, 6} are disjoint sets, while D= {1, 2, 3} and E={3, 4, 5} are not.

Power Set:

The collection of all subsets of set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.

For example;

If A = {p, q} then all the subsets of A will be ∅, {p}, {q} and {p, q}. And the power set of A is P(A) = {∅, {p}, {q}, {p, q}}.

Number of elements of P(A) = n[P(A)] = 4 = 22.

Generally, n[P(A)] = 2m where m is the number of members in set A.

Family of sets/Sets of sets/Class of sets:

It sometimes will happen that the objects of a set are set themselves; for example, the set of all the subset of A; that is

A={{2,3},{2},{5,6} is a family of sets. Its members are the sets {2,3},{2} and {5,6}.

Universal Set

A set that contains all the elements of other given sets is called a universal set or universe of discourse.. The symbol for denoting a universal set is .

For example;

  1. A = {All the people of the world}
  2. B={All the points of the plane}
  3. If P is a set of all whole positive numbers and Q is a set of all negative numbers then the universal set is a set of all integers.

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