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We can extend the concept of subtraction, used in algebra, to the sets. If a set “B” is subtracted from set “A”, the resulting difference set consists of elements, which are exclusive to set “A”. We represent the symbol of difference of sets as “A-B” and pronounce the same as “A minus B”.

Difference of sets
The difference of sets “A-B” is the set of all elements of “A”, which do not belong to “B”.

In the set builder form, the difference set is :

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


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

On Venn’s diagram, the difference "A-B" is the region of “A”, which excludes the common region with set “B”.

Difference of two sets

The difference of two sets is a disjoint set.

Interpretation of difference 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 difference 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 can summarize two statements with two ways arrow as :

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

Composition of a set

From Venn’s diagram, we observe that if we derive union of ( A B ) to either of the difference sets, then we get the complete individual set.

Difference of two sets

The difference of two sets is a disjoint set.

A = A - B A B


B = B - A A B

Difference of sets is not commutative

The positions of sets about minus operator affect the result. It is clear from the figure above, where “A-B” and “B-A” represent different regions on Venn’s diagram. As such, the difference of sets is not commutative. Let us consider the example used earlier, where :

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

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


A B = { 1,2,3 }


B A = { 7,8 }



Symmetric difference

From the Venn’s diagram, we can see that union of two sets is equal to three distinct regions. Alternatively, we can say that the region represented by the union of two sets is equal to the sum of the regions representing three “disjoint” sets (i) difference set A-B (ii) intersection set " A B " and (iii) difference set B-A.

Difference of two sets

The difference of two sets is a disjoint set.

We use the term “symmetric set” for combining two differences as marked on Venn’s diagram. It is denoted as “ A Δ B ”.

A Δ B = A B B A

Complement of a set

The complement is a special case of the difference operation. The set in question is subtracted from universal set, “U”. Thus, one of the sets in difference operation is fixed. We define complement of a set as its difference with universal set, "U". The complement of a set is denoted by the same symbol as that of set, but with an apostrophe. Hence, complement of set A is set A’.

Complement of a set
The complement of a set “A” consists of elements, which are elements of “U”, but not the elements of “A”.

We write the complement set in terms of set builder form as :

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

Note that elements of A’ does not belong to set “A”. On Venn’s diagram, the complement of “A” is the remaining region of the universal set.

Complement of a set

The complement of a set is the remaining region of the universal set.

Questions & Answers

I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
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How can I make nanorobot?
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
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Period of sin^6 3x+ cos^6 3x
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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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