SET OPERATION

Set Operations.

The Process of making a new sets from two or more given sets applying some special rules is known as set operations.

If we are given two sets , then there are three standard ways to construct new sets from them. The three operations are called binary set operations , which are as following:

Union:

A set that contains all the elements contained by first set (A) and second set (B) is known as union of the two sets (A and B).

We denote union of two sets (A and B) by symbol A ∪ B.

For example: if A={1,2,3} and B={3,4,5} Then,
A ∪ B={1,2,3,4,5}

Intersection:

A set whose elements are the common elements of two sets (A and B) is known as the intersection of the sets(A and B).  The intersection of two sets (A and B) is denoted by the symbol A ∩ B.

For example: If A={1,2,3} and B={2,3,4} Then A∩B={2,3}

Complement:

A set whose elements are all the elements of universal set except a set (A)is known as the complement of the set (A). The complement of a set (A) is denoted by symbol  and read as “A complement”

For example: If A={1,2,3} , B={3,4,5} and C={4,5,6,7} Then , Â={4,5,6,7}

Difference:

The difference of set A and B is the set formed by a set with all elements of set A that does not belongs to set B. We denote the difference of set A and B by A-B and difference if set B and A by B-A.

For example: If As={1,2,3,4} and B={3,4,5,6} then,

A-B={1,2} , B-A={5,6} , A-A= φ and B-B= φ

Above set operations are shown below as graphical representation in Venn diagram.

Union:

In the following figures A∪B is shown as shaded region:

Intersection:

In the following figures A∩B is shown as shaded region , in second figure no region is shaded because in the figure A∩B=Φ

Complement:

In the folowing figure  is shown by shaded region:

Difference:

In the first figure below A-B is shown as shaded region and in second figure A-A is shown and no region as shaded as A-A is Φ

The concept of modern mathematics is started with set. Set appears in all branches of mathematics. The main developers of set theory is George Cantor (1845-1915 AD) and presented by zakayo Mong’ateko .

                                 

George Cantor                                    zakayo Mong’ateko

The word set is synonym with “Collection” , “Class” or “Aggregate”. Basically set is a collection or organization of similar objects (an object may be material or conceptual). and the objects by which a set is made of  are called elements or member of the set. Some example of set are:
1> The countries of Europe.

2>The solar system.

3>The peoples living in my house.

4>Vowels of the English alphabets. etc.

Notation:

Sets are usually denoted by capital letters and the elements of set are denoted by small letters. For example : Set A={a,b,c,d} The symbol    denotes set membership and symbol    denotes non membership. For example in set A above, a   A but e  A , Which means the element “a” belongs to set “A” but the “e” doesn’t.

Specification:

A set can be denoted or membership of a set may be indicated in several ways. Two common ways among them are:

a> Listing or Tebulation. In this method elements of set are listed , sperated by commas and enclosed inside a bracket. For example: V={a,e,i,o,u} My family={me , my son , my spouse}

b>Description or Rule. In this method a set is specified by enclosing in brackets a descriptive phrase or a rule. For example: V={the vowels of the English alphabet.} N={x:x is a natural number} A={x:x^2-3x=0} In last two examples the respective set is a group of element x , where each value of element x is defined by the respective rule.

Finite and Infinite set:

If a set have a finite numbers of elements then a set is called finite set else the set is called infinite set. For example: D={0,2,4,6,8} A={a,e,i,o,u} The above two sets are finite set while the following sets are infinite one. The set of stars in Universe. D={set of natural numbers}

Null Set:

A set which has no element is called Null set. A Null set is also called Empty set or Void set. It is denoted by symbol ø  For example: A={x:x is a man who gave birth to a child) N={x:xis a natural number , x>0 and x<1 }

To learn more about set theory please browse through our site

Relation between sets.

If , in any condition two or more sets appears in discussion they might have some special relation between each other. There are many types or relation that might occur between two or more sets. Those relations are:

Subset:

If one set (A) contains  all the elements that another set (B) contains then the second set (B) is called to be the subset of first set (A) , or set B contains set A.

In symbol we write

A ⊂ B  (A is contained in B) , B ⊃ A (B contains A)

Both symbols above means that set A is a subset of set B.

A set may have two or more subsets.

For example: If set A={1,2,3,4} Then {1} , {2,3} , {4,1} etc. are the subsets of set A.

Note:

*The number of elements a set contains is known as its cardinal number.

*The number of possible subset a set can have is given by the formula 2^s , Where “s” is the cardinal  number  of set A.

*If a set contains all other set that are currently being discussed then the set is called universal set.

Equal set:

Two sets are said to be equal if every element contained by first set is contained by second set and also every elements contained by second set is contained by first set.

equal sets are  sub set of each other.

For example:

If set A={a,b,c,d} and set B={d,a,b,c} Then set A and set B are equal set.

Proper Subset:

If A ⊂ B and A ≠ B(A is not equal to B) then set A is said to be a proper subset of set B. In other words , A set is said to proper subset of another set if every elements contains by the set in contained by another also but the another one also contains some elements not contained in the first set.

For Example:  If A={1,2,3} and B={1,2,3,4} then set A is a proper subset of set B.

Power set:

A set of all subsets of any set is known as power set. It is denoted by “2^s”.

For example:  If S={a,b} then all possible subsets of set S are :  ø , {a} , {b} ,{a,b}

So , 2^s of set S is [ø , {a} , {b} ,{a,b}]

As told on the note above the cardinal number of  power set is given by formula 2^s where “s” is the cardinal number of any set.

Disjoint sets:

Two sets are said to be Disjoint if they dont have any common element.

For example: The set of boys and the set of girls is disjoint , If set A={1,2} and set B={3,4} then set A and B are disjoint.

Intersecting sets:

Two sets are said to be intersecting if some of elements they have are common in both.

For example: If set A={1,2,3} and set B={3,4,5} then set A and set b are  intersecting sets.
Definition:

Given two sets A and B, the intersection is the set that contains elements or objects that belong to A and to B at the same time

We write A Ç B

Basically, we find A Ç B by looking for all the elements A and B have in common. We next illustrate with examples


Example #1.


To make it easy, notice that what they have in common is in bold

Let A = {1 orange, 1 pinapple, 1 banana, 1 apple} and B = { 1 spoon, 1 orange, 1 knife, 1 fork, 1 apple}

A Ç B = {1 orange, 1 apple}


Example #2.


Find the intersection of A and B and then make a Venn diagrams.

A = {b, 1, 2, 4, 6} and B = { 4, a, b, c, d, f}

A Ç B = {4, b}

Example #3.

A = { x / x is a number bigger than 4 and smaller than 8}

B = { x / x is a positive number smaller than 7}

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

A Ç B = {5, 6}

Or A Ç B = { x / x is a number bigger than 4 and smaller than 7}

Example #4.


A = { x / x is a country in Asia}

B = { x / x is a country in Africa}

Since no countries in Asia and Africa are the same, the intersection is empty

A Ç B = { }

Example #5.

A = {#, %, &, *, $ }

B = { }

This example is subtle! Since the empty set is included in any set, it is also included in A although you don't see it

Therefore, the empty set is the only thing set A and set B have in common

A Ç B = { }


In fact, since the empty set is included in any set, the intersection of the empty set with any set is the empty set.

Definition of the union of three sets:

Given three sets A, B, and C the intersection is the set that contains elements or objects that belong to A, B, and to C at the same time

We write A Ç B Ç C

Basically, we find A Ç B Ç C by looking for all the elements A, B, and C have in common.


A = {#, 1, 2, 4, 6}, B = {#, a, b, 4, c,} and C = A = {#, %, &, *, $, 4 }

A Ç B Ç C = {4 , # }

The graph below shows the shaded region for the intersection of two sets

The graph below shows the shaded region for the intersection of three sets

This ends the lesson about intersection of sets. If you have any questions about t

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                               Mwl. Zakayo Mong’ateko-