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  • elements exclusive to A
  • elements exclusive to B
  • elements common to A and B

As a set includes only distinct elements, the common elements are represented only once in the union set. Thus, union set consists of elements of both sets without repeating an element. Now, the set is represented on Venn diagram as shown here.

Union of two sets

The representation of union on Venn diagram

For illustration of working with union, let us consider two sets of positive integers as given here,

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

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

The union of two sets is :

A B = { 1,2,3,4,5,6,4,5,6,7,8 }

But repetition of elements in a set does not change it. Hence, we need not repeat elements in the resulting union.

A B = { 1,2,3,4,5,6,7,8 }

Here, universal set is natural numbers. The representation of union of joint sets is shown in the figure. We can observe that very construction of union on Venn diagram ensures that elements are not repeated.

Union of two sets

The representation of union on Venn's diagram

Interpretation of union set

Let us examine the defining set of union :

A B = { x : x A o r x B }

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

I f x A B , t h e n x A o r x B .

Can we emphasize this conditional meaning in reverse order :

I f x A o r x B , t h e n x A B .

Yes, we can agree with the second conditional meaning as well. We, therefore, conclude that the statements work in both ways. We write two statements together as :

x A B x A o r x B

We can reach yet another conclusion by observing representation of union set on Venn diagram. Now, if an arbitrary element “x” does not belong to union set, then it is clear that it does not belong to the region represented by the union set on the Venn’s diagram. Hence,

Union of two sets

The representation of union on Venn's diagram

I f x does not belong A B x A and x B .

The important thing to note here is the word “and” in place of “or” used before. Think about it. Here two conditions follow simultaneously. If an element does not belong to an union set, then it will not belong to either of individual sets simultaneously. Now, the next thing to consider is whether this conditional statement will be true other way round as well?

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

Yes, we can agree to this statement. We, therefore, conclude that the statements work in both ways. We write two statements together with the help of two ways arrow sign as :

x A B x A and x B .

Union of disjoint sets

Consider students in class X and class XI. Let us denote the respective sets as "T" for tenth and "E" for eleventh class. Clearly, union i.e. combination of two sets should include elements from each of the sets. Hence,

T E = students in class X and XI

This is a straight forward union of two sets. The resulting set comprises of all elements present in both the sets. Since it is not possible that students studying in class X are also students of XI, we are sure that the numbers of elements in the union is sum of numbers of students in each class. As there is no commonality between two sets, it is a union of two “disjoint” sets. We conclude here that union of two disjoint sets has no common elements.

Union with subset

The set “B” consists of all elements of its subset “A”. In other words, the elements of a subset “A” also belongs to the set “B”. The operation of union is combining elements of two sets. The union with a subset, therefore, consists of elements from both “A” and “B”. However, all elements of “A” are also the elements of “B”. Therefore, we find that union set is same as the superset “B”. Symbolically,

I f A B , then B A = B .

We can check this deduction with the help of an example. Let us consider two sets as :

A = { 4,5,6 }

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

Here, we see that A B. Now,

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

Union with subset

The representation of union with subset on Venn diagram

Multiple unions

If A 1 , A 2 , A 3 , , A n is a finite family of sets, then their unions, 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 union operation.

Idempotent law

The literal meaning of the word “idempotent” is “unchanged when multiplied by itself”. Following the clue, the union of a set with itself is the set itself. This is an equivalent statement conveying the meaning of “idempotent” in the context of union. Symbolically,

A A = A

The union set consists of distinct elements and common elements taken once. Between two sets here, all elements are common. The union set consists of all elements of either set.

Identity law

The algebraic operators like addition and multiplication have defined identities, which does not change the other operand of the operator. For example, if we add “0” to a number, it remains same. Hence, “0” is additive identity. Similarly, “1” is multiplicative identity.

In the case of union, we find that union of a set with empty set does not change the set. Hence, empty set is union identity.

A φ = A

As there is no element in empty set, union has same elements as that in “A”.

Law of u

All sets are subsets of universal set for a given context. We have seen that union with subset results in the set itself. Clearly, union of universal set with its subset will result in the universal set itself.

U A = U

Commutative law

In order to assess whether commutative property holds or not, we consider the example, used earlier. Let the sets be :

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

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

Then,

A B = { 1,2,3,4,5,6,4,5,6,7,8 } = { 1,2,3,4,5,6,7,8 }

B A = { 4,5,6,7,8,1,2,3,4,5,6 } = { 1,2,3,4,5,6,7,8 }

Thus, we see that order of operands with respect to the union operator is not differentiating. We can also appreciate this law on Venn diagram, which does not change by changing positions of sets across union operator.

Associative law

The associative property also holds with respect to union operator. We know that associative property is about changing the place of parentheses as here :

A B C = A B C

The parentheses simply change the precedence of operation. On Venn diagram, union involving three sets appears same, irrespective of whether we apply union operation in a particular sequence.

Union of three sets

Associative law

Questions & Answers

how do you translate this in Algebraic Expressions
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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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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
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
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
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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