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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 }

and

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

and

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 }

Then,

A B = { 1,2,3 }

and

B A = { 7,8 }

Clearly,

A B B A

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

how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
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what is nano technology
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
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AMJAD
what is the application of nanotechnology?
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In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
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not now but maybe in future only AgNP maybe any other nanomaterials
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can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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
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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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