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Basic Operations on Sets

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One of the main applications of Sets theory is to make the Relations between two sets. For making these relations, certain operations can be performed on the Numbers. We will discuss Some Basic Operations on Sets.

Union: As the word indicates Union means joining or adding together. Union of Two sets means adding together the two sets.  Let us consider two sets ‘A’ and ‘B’, then the union of two sets is represented by “A U B”, which means all the elements which either belong to ‘A’ or ‘B’. Here are some examples: If A= 1, 2, 3 and B= Red, Blue

So we have A U B = 1, 2, Red, Blue

 Again if A = 1, 3, 5 and B =   2, 3, 4, 6.

 So we have A U B = 1, 3, 5, 2, 4, 6.

 Also if A= 1, 2 and B = 1, 2.

So we have A U B = 1, 2.

Following are some properties of Union of sets:

a)     A U B = B U A, this property is known as Commutative Property.
b)     A U ( B U C ) = ( A U B ) U C, this property is known as Associative Property.
c)      A U A = A,
d)     A U φ = A,
e)     ‘A’  is the subset of ‘B’ if A U B = B.

Intersection: We can also form a new Set by determining the common elements of the given two sets. This is called Intersection of the two sets.

The intersection of 'A' and 'B', denoted by A$\cap$B

 If A $\cap$ B =φ , then 'A' and 'B' are  called disjoint sets. 

Union

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Set can be defined as the collection of well defined and distinct objects, considered as an object. While studying Sets theory, we need to reform the group of the sets in order to get the solutions. One of the main applications of the sets theory is to make the Relations between two sets. For making these relations, certain operations can be performed on the Numbers. We will discuss these operations one by one.
Union: As the word indicates union means joining or adding together. Union of sets means adding two sets. Let us consider two sets A and B, then the union of sets is represented by  A U B, which means all the elements, which either belong to A or B. Here are some examples: If A= 1, 2, 3 and B= table, chair. So, we have A U B = 1, 2, table, chair.
Thus we conclude that if the two sets are disjoint, then  union Set will have the set with all the elements of Set A and Set B.
In another example if A = 1, 3, 5 and B = 1, 2, 3, 4, 5, 6 then we have A U B = 1, 2, 3, 4, 5, 6. Here we find that A U B will represent all the elements belonging to set A and set B.
Here we have some common properties of union of sets:
a)     A U B = B U A.
b)     A U (B U C) = (A U B) U C.
c)      A U A = A.
d)     A U φ = A.
e)     A is the subset of B if A U B = B.

Complements

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A Set can be defined as a well organized collection of distinct objects, placed under a group. We sometimes say that a set is the most elementary concept studied in mathematics.
Different mathematical operations can be performed on Sets. When we talk about the complement of a set, it simply means that we want to represent the elements of a universal set, which are not in the given set.
Let us consider ‘U’ as a universal set and let ‘A’ be any subset of set ‘U’. So ‘A’ represents a complement set, which has all the elements of set ‘U’, which are not in set ‘A’.
 It can be mathematically represented as, A’ = U – A. We can say that complement of set ‘A’ in ‘U'. So
we can say that in a given set ‘A’, the complement of ‘A’ is the set of all element in the universal set ‘U’, but not in ‘A’
We can also say that the complement of set ‘A’ in ‘U’, let us take an example to understand above concept-
 Suppose we have a set ‘B’, this set contain fruits in a basket  which are, one each of orange, pineapple , banana and apple. It is represented as:
Therefore B = 1 orange, 1 pineapple, 1 banana, 1 apple .
 Now let the universal set be ‘U’, such that it contains following fruits-
Let U = 1 orange, 1 apricot, 1 pineapple, 1 banana, 1 mango, 1 apple, 1 papaya. 
So we need to find the fruits which are in universal set but not in set ‘B’, it will be represented by B’ and we call it Complement of set ‘B’.
B = 1 apricot, 1 mango, 1 papaya.

Cartesian Products

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The Cartesian product of two Sets is defined in terms of ordered pairs. The Cartesian product sets are also called cross join sets in which the product of each row of the first table multiplied by each column of another table. Such situation arises when their exist no relation between the two given tables.
The Cartesian product of sets A and B is defined as the Set of all the points (a, b), where a belongs to A and b belongs to B and the Cartesian product is denoted by A X B. The set so produced is also called the product set or the cross product. A Cartesian product of sets is a construction to build a new set from the number of given sets.
The Cartesian product of two sets X (for example the points on an x-axis) and Y (for example the points on a y-axis), denoted X × Y, is the set of all possible pairs of elements  whose first component is a member of set  X and whose second component is a member of set Y. If we consider the elements of X-Y plane, then any Point we take on XY plane is represented as (x, y). A Cartesian product of any given two finite sets can be represented by a table, forms the ordered pairs, the first cell of which is the element of first set and the second cell is the element of the second set.
If the two sets are finite, then the number of elements in the Cartesian product is also finite. The Cartesian product of a non-empty set with an empty set is equal to empty set. So we can say if A is any finite set then A * φ = φ.  
On the other hand, if one of the sets is infinite, then we get resulting Cartesian product also as infinite. The Cartesian Product of finite set can be expressed as A * B = (x, y): x Є A, y Є B.

Intersection

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We can say that a Set is the collection of different objects grouped together. Any set is the most basic element used in mathematics. In everyday life we observe that Sets are used to make groups of same type of elements. When we study sets theory, we need to re- organize the group of the sets which always helps us to get the solutions. One of the main applications of the sets theory is to make the Relations between two sets and then observe the elements of the new set. For making these relations, certain operations can be performed on the Numbers. We will discuss these operations one by one.
Intersection of two sets:  We can also form a new set by determining the common elements of the given two sets. This is called intersection of the two sets.  If A∩ B = ∅, then ‘A’ and ‘B’ are said as disjoint setsIntersection of a set is a basic operation performed in the sets Algebra. Here we must first know that as Union represents “ OR ” operation, similarly  Intersection represents “AND” operation, which means the element exist in both the sets,  will belong to the intersection sets.  It means that in order to form the new set with the  intersection sets ( A and B ), the new set will have to coincide in both or more sets. 
Here are some of the examples to show the intersection of sets. Let A= 1, 2 and B= red, black then ‘A’ intersection B = φ, Null set. We observe that there is no element common in both the sets, so the result is the null set (a set with no element).
Similarly if A= 1, 2, 3, 4 and B = 4, 5, 6, 7.
 Here both sets ‘A’ and ‘B’ have 4 as common, so ‘A’ intersection ‘B’ will be = 4
Again we observe that if set ‘A’ and set ‘B’ have all elements same, then ‘A’ intersection B = A.