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We have pointed out that a set representing a real situation is not an isolated collection. Sets, in general, overlaps with each other. It is primarily because a set is defined on few characteristics, whereas elements generally can possess many characteristics. Unlike union, which includes all elements from two sets, the intersection between two sets includes only common elements.

Intersection of two sets
The intersection of sets “A” and “B” is the set of all elements common to both “A” and “B”.

The use of word “and” between two sets in defining an intersection is quite significant. Compare it with the definition of union. We used the word “or” between two sets. Pondering on these two words, while deciding membership of union or intersection, is helpful in application situation.

The intersection operation is denoted by the symbol, " ". We can write intersection in set builder form as :

Intersection of two sets

The intersection set consists of elements common to two sets.

A B = { x : x A a n d x B }

Again note use of the word “and” in set builder qualification. We can read this as “x” is an element, which belongs to set “A” and set “B”. Hence, it means that “x” belongs to both “A” and “B”.

In order to understand the operation, let us consider the earlier example again,

A = { 1,2,3,4,5,6 }

B = { 4,5,6,7,8 }

Then,

A B = { 4,5,6 }

On Venn diagram, an intersection is the region intersected by circles, which represent two sets.

Intersection of two sets

The intersection set consists of elements common to two sets.

Interpretation of intersection set

Let us examine the defining set of intersection :

A B = { x : x A a n d x B }

We consider an arbitrary element, say “x”, of the intersection set. Then, we interpret the conditional meaning as :

I f x A B x A a n d x B .

The conditional statement is true in opposite direction as well. Hence,

I f x A a n d x B x A B .

We summarize two statements with two ways arrow as :

x A B x A a n d x B

In addition to two ways relation, there is an interesting aspect of intersection. Intersection is subset of either of two sets. From Venn diagram, it is clear that :

Intersection of two sets

The intersection set consists of elements common to two sets.

A B A

and

A B B

Intersection with a subset

Since all elements of a subset is present in the set, it emerges that intersection with subset is subset. Hence, if “A” is subset of set “B”, then :

B A = A

Intersection of disjoint sets

If no element is common to two sets “A” and “B” , then the resulting intersection is an empty set :

A B = φ

In that case, two sets “A” and “B” are “disjoint” sets.

Multiple intersections

If A 1 , A 2 , A 3 , , A n is a finite family of sets, then their intersections one after another is denoted as :

A 1 A 2 A 3 . A n

Important results

In this section we shall discuss some of the important characteristics/ deductions for the intersection operation.

Idempotent law

The intersection of a set with itself is the set itself.

A A = A

This is because intersection is a set of common elements. Here, all elements of a set is common with itself. The resulting intersection, therefore, is set itself.

Identity law

The intersection with universal set yields the set itself. Hence, universal set functions as the identity of the intersection operator.

A U = A

It is easy to interpret this law. Only the elements in "A" are common to universal set. Hence, intersection, being the set of common elements, is set "A".

Law of empty set

Since empty set is element of all other sets, it emerges that intersection of an empty set with any set is an empty set (empty set is only common element between two sets).

φ A = φ

Commutative law

The order of sets around intersection operator does not change the intersection. Hence, commutative property holds in the case of intersection operation.

A B = B A

Associative law

The associative property holds with respect to intersection operator.

A B C = A B C

The intersection of sets “A” and “B” on Venn’s diagram is :

Intersection of two sets

The intersection is a set of common elements and shown as colored region.

In turn, the intersection of set “A B” and set “C” is the small region in the center :

Intersection inloving three sets

Intersection of a set with "the intersection set of two sets"

It is easy to visualize that the ultimate intersection is independent of the sequence of operation.

Distributive law

The intersection operator( ) is distributed over union operator ( ) :

A B C = A B A C

We can check out this relation with the help of Venn diagram. For convenience, we have not shown the universal set. In the first diagram on the left, the colored region shows the union of sets “B” and “C” ie. B C . The colored region in the second diagram on the right shows the intersection of set “A” with the union obtained in the first diagram i.e. B C .

Distributive law

Distribution of intersection operator over union operator

We can now interpret the colored region in the second diagram from the point of view of expression on the right hand side of the equation :

A B C = A B A C

The colored region is indeed the union of two intersections : " A B " and " A C " . Thus, we conclude that distributive property holds for "intersection operator over union operator".

In the same manner, we can prove distribution of “union operator over intersection operator” :

A B C = A B A C

Analytical proof

Distributive properties are important and used for practical application. In this section, we shall prove the same in analytical manner. For this, let us consider an arbitrary element “x”, which belongs to set " A B C " :

x A B C

Then, by definition of intersection :

x A a n d x B C

x A a n d x B o r x C

x A a n d x B o r x A a n d x C

x A B o r x A C

x A B o r A C

x A B A C

But, we had started with " A B C " and used its definition to show that “x” belongs to another set. It means that the other set consists of the elements of the first set – at the least. Thus,

A B C A B A C

Similarly, we can start with " A B A C " and reach the conclusion that :

A B A C A B C

If sets are subsets of each other, then they are equal. Hence,

A B C = A B A C

Proceeding in the same manner, we can also prove other distributive property of “union operator over intersection operator” :

A B C = A B A C

Questions & Answers

Differences between microeconomics and macroeconomics
tatiana Reply
what is Economics
Ebem Reply
the branch of knowledge concerned with the production, consumption, and transfer of wealth and has Influence by sociology!!!!
Ajay
Economics is the study of how humans make decisions when they want to fulfil their requirements and desires for goods, services and resources.
Abdullah
Economics is the study how humans make decisions in the faces of scarcity.
Rose
economic is the study of how human make decision in the fact of scarcity.
Toang
Economics is a social science which study human behavior as a relationship between earn and scarce mean which have alternative uses
Juliet
what is market structure
Fatima
market structure in economics depicts how firms are differentiated and categorised based on types of goods they sell and how their operations are affected by external factors and elements.
Nasir
what is economic theory
Madara
what is demand
Gooluck Reply
demand is the willingness to purchase something
Mohamed
demand is the potential ability or williness to purchases something at a particular price at a given period of time..
Ahmed
Demand refers to as quantities of a goods and services in which consumers are willing and able to purchase at a given period of time. Demand can also be defined as the desire backed by ability to purchase .
Fadiga
what is demand
John Reply
is the production of goods in scarcity
David
thanks
John
Demand refers to as quantities of a goods and services in which consumers are willing and able to purchase at a given period of time.
Fadiga
what is demand of supply
music Reply
What is the meaning of supply of labour
Anthonia Reply
what is production?
Elizabeth Reply
Production is basically the creation of goods and services to satisfy human wants
Anthonia
under what condition will demand curve slope upward from left to right instead of normally sloping downward from left to right
Atama Reply
how i can calculate elasticity?
Tewekel Reply
What is real wages
Emmanuella Reply
what are the concept of cost
Tabitha Reply
what is the difference between want and choice
Grace Reply
Want is a desire to have something while choice is the ability to select or choose a perticular good or services you desire to have at a perticular point in time.
Dalton
substitutes and complements
Amman Reply
Substitute are goods that can replace another good but complements goods that can be combined together
nkanyiso
account for persistent increase in lnflation
niwahereza Reply
what is opportunity cost
Adebowale Reply
opportunity cost reffered to as alternative foregone when choice is made
niwahereza
Difference between extinct and extici spicies
Amanpreet Reply
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah Reply
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Period of sin^6 3x+ cos^6 3x
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Period of sin^6 3x+ cos^6 3x
Sneha Reply

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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