Type of Set

Empty Set : 

• Definition : a  set contains no element
• Denote by ∅ or { }

Figure above shows there is no element in an empty set

Example :
a) The set of rational numbers which has reminder.

• Clearly , there is no rational number with reminder. Therefore, it is an empty set.

b) The set of even prime numbers greater than 2

• prime number does not contain even number in it. Therefore , it is an empty set

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Set equality :

• Definition : Two or more sets are equal if and only if they have the same elements. 
• Two sets are not equal if they do not have the same elements.

A = {1,3,7}   B = {1,3,3,3,7,7,7}   C = {7,1,3}

Based on the figure above, set A, set B and set C are equal sets because these three sets have the same element which is 1, 3 and 7 even though the order of elements in set C is not the same as set A and the amount of elements in set B is higher than that in set A and set B respectively.

We can conclude that A = B = C

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Subset :

• Definition : a set of elements that are also in another set.

           Y = {1,2,3,4,5,6,7,8,9,10}

           Z = {2,4,6,8,10}

• Set Z is said to be a subset of set Y if and only if all elements of the set Z are included in the set Y.  //  Z ⊂ Y
• Symbols for subset are ⊂, ⊃, ⊄, ⊅, ⊆, ⊇, ⊈ and ⊉. 

The examples of different types of subset are stated in the table below:

Symbol for subset	Symbol name	Meaning / definition	Example
X ⊂ Y	Proper subset	Set X has some elements of set Y.	{1, 2, 3} ⊂ {1, 2, 3, 4}
X ⊃ Y	Proper superset	Set X has set Y’s elements and more	{1, 2, 3, 4} ⊃ {1, 2, 3}
X ⊆ Y	Subset	Set X has some or all elements of set Y	{4, 5, 6} ⊆ {4, 5, 6}
X ⊇ Y	Superset	Set X has same elements as set Y, or more	{4, 5, 6} ⊇ {4, 5, 6}
X ⊄ Y	Not a subset	Set X is not a subset of set Y	{1, 9} ⊄ {1, 2, 3, 4}
X ⊅ Y	Not a superset	Set X is not a superset of set Y	{1, 2, 3, 4} ⊅ {1, 9}

Based on the figure above, set A is a subset of set B because all the elements in the set A are included in the set B.

Reference :

• https://en.wikipedia.org/wiki/Subset#%E2%8A%82_and_%E2%8A%83_symbols

• https://www.rapidtables.com/math/symbols/Set_Symbols.html 

• https://www.mathsisfun.com/sets/symbols.html

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Power Set :

• Definition : a set of all the subsets of a set.
• The symbol for represent a set (set A for example) is P(A).

Example: for the set S={2,4,5} how many members will the power set have?

There are two method we can use to determine power set :

1. Expand the element by multiplying element respectively 

• the empty set, {} is a subset of set S.
• these are subsets: {2}, {4}, {5}, {2, 4}, {2, 5} and {4, 5}
• {2, 4, 5} is also a subset of {2, 4, 5}.
• therefore, the power set of set A is P(S) = {{ },{1},{2},{3},{1, 2},{1, 3},{2, 3},{1, 2, 3}}

Answer :  P(S) = 8

 2.The number of members of a set is often written as |S|, so when S has n members we can write:

|P(S)| = 2^n

|P(S)| = 2^n  = 2^3 = 8

Answer : P(S) = 8

Reference : https://en.wikipedia.org/wiki/Power_set

         
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Union Set :

• Definition :  union of two given sets is the smallest set which contains all the elements of both the sets.
• The union of two sets can be found by combining all elements of two sets.
• The union of two sets A and B has the meaning of “A or B”.
• For example: Given that the set A = {1, 2, 3} and set B = {a, b, c}. The union of two sets A and B is
• A ∪ B = {1, 2, 3, a, b, c}

The figure for this example is as follow:

Reference :  https://www.mathgoodies.com/lessons/sets/union

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Intersection Set : 

• Definition : The intersection of two sets A and B (denoted by A ∩ B), is the set of all objects that are members of both the sets A and B.
• The intersection of two sets can be found by finding elements in common to both sets.
• The intersection of two sets A and B has the meaning of “A and B”.
• For example: Given that set A = {1, 2, 3, 4, 5} and set B = {3, 4, 5, 6, 7}. The intersection of sets A and B is :

                                          A ∩ B = {3, 4, 5}

The figure for this example below is as follow : 

Reference : https://brilliant.org/wiki/sets-union-and-intersection-easy/ 

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Disjoint Set : 

• Definition : Two sets are called disjoint if their intersection is an empty set.
• For example: Given that set A = {1, 3, 5, 7} and set B = {2, 4, 6, 8}. The intersection of sets A and B is

                                       A ∩ B = {} = Ø

• Therefore, sets A and B are disjoint.

The figure for this example is as follow : 

Reference : http://mathworld.wolfram.com/DisjointSets.html

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Set Difference :

• The difference of sets A and B, A – B is also called as the complement of B with respect to A.
• For example: Given that set A = {a, b, c} and set B = {b, c, d}. The difference of sets A and B is :

                                                 A – B = {a}, but B – A = {d}

• The figures for this example is as follow : 

Reference : https://proofwiki.org/wiki/Definition:Set_Difference

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Set complement :

• Definition :  the complement of a set A refers to elements not in A.
• If A and B are sets, then the set complement of A in B, also termed the set difference of B and A, is the set of elements in B but not in A.
• If A is a set, then the symbols for set complement of set A are Ac, A’, Ā, CvA and C.
• For example: Given that universal set, U = {1, 2, 3, 4, 5, 6, 7} and set A = {1, 2, 3}. The complement of A with respect to U is :

                      

    Ac = {4, 5, 6, 7}

The figure for this example is :

In the figure above, the red part is the set complement, Ac.

Reference :  https://en.wikipedia.org/wiki/Complement_(set_theory)

Properties of Set

1. The change in order of writing the elements does not make any changes in the set.

If we change the order of the element in a set it will not change anything.
In other words the order in which the elements of a set are written is not important. Thus, the set {a1, b1, c1} can also be written as {a1, c1, b1} or {b1, c1, a1} or {b1, a1, c1} or {c1, a1, b1} or {c1, b1, a1}.

For Example:

Set A = { 6, 7, 8, 9} is same as set A = {8, 9, 7, 6}

i.e., { 6, 7, 8, 9} = {8, 9, 7, 6}

Similarly, {w, x, y, z} = {x, z, w, y} = {z, w, x, y}    and so on.






2. If one or many elements of a set are repeated, the set remains the same.

In other words the elements of a set should be distinct. So, if any element of a set is repeated number of times in the set, we consider it as a single element. Thus, {1, 1, 2, 2, 3, 3, 4, 4, 4} = {1, 2, 3, 4}

The set of letters in the word ‘YODOO’ = {Y, O, D }

For Example:

The set A = {5, 6, 7, 6, 8, 5, 9} is same as set A= {5, 6, 7, 8, 9}

i.e., {5, 6, 7, 6, 8, 5, 9} = {5, 6, 7, 8, 9}

In general, the elements of a set are not repeated. Thus,

if T is a set of letters of the word ‘moon’: then T = {m, o, n},

There are two o’s in the word ‘moon’ but it is written in the set only once.

Reference : https://math.stackexchange.com/questions/2366063/prove-properties-of-sets-and-powersets

Element of Set

The objects used to form a set are called its element or its members.

Generally, the elements of a set are written inside a pair of curly (idle) braces and are represented by commas. The name of the set is always written in capital letter.

Example :
 If set S = {teacher, student, staff, librarian, gardener}. State which of the following statements are ‘correct’ and which are ‘wrong’ and explain the statement. (S = School)

(i)  bag ∈ S

(ii)  teacher ∈ S

(iii) student ∈ S

(iv) pencil box ∈ S

Solution:

(i) bag ∈ S

Wrong, since bag does not belongs to  set S i.e. bag ∉ S

(ii) teacher ∈ S

Correct, since teacher belongs to  set S.

(iii) student ∈ S

Correct, since student belongs to  set S.

(iv) pencil box ∈ S

Wrong, since pencil box does not belongs to set S i.e. pencil box ∉ S

Reference : http://www.mathwords.com/e/element_of_a_set.htm

Set Theory

Definition of Set :

A set is an unordered collection of objects. We might also consume a set is a collection of object called element. 

Example :

In the circle above consist of a set of different shape of polygons. The polygons are the element of the set 

Other example are :
i.The students in this class
   >Students are the element in the set of class.
ii. People in office
   >People are the element in the set of office.


Reference : https://www.mathsisfun.com/definitions/set.html