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Interpretation of complement

Proceeding as before we can read the conditional statement for the complement with the help of two ways arrow as :

x A x U a n d x A

In terms of minus or difference operation,

A = U A

It is clear from the representation on Venn’s diagram that the universal set comprises of two distinct sets – set A and complement set A’.

U = A A

Compliment of universal set

The complement of universal set is empty set. It is so because difference of union set with itself is the empty set (see Venn's diagram).

U = { x : x U a n d x U } = φ

Complement of empty set

The complement of the empty set is universal set. It is so because difference of union set with the empty set is universal set (see Venn's diagram).

φ = { x : x U a n d x φ } = U

Complement of complement set is set itself

The complement of complement set is set itself. The complement set is defined as :

A = U A

Now, complement of complement set is :

A = U A

Let us consider the example, where :

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

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

Then,

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

Again taking complement, we have :

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

Union with complement set

The union of a set with its complement is universal set :

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

From Venn’s diagram also, we see that universal set consists of set A and component A’.

U = A A

The two sets on the right side of the equation are disjoint sets. Hence,

A A = U

Intersection with complement set

There is nothing common between set A and its component A’. Thus, intersection of a set with its complement yields the empty set,

A A = φ

De-morgan’s laws

In the real world situation, we want to negate a condition of incidence. For example, consider a class in the school. Some students play either basketball or football or both, but there are students, who play neither basketball nor football. We have to identify later category of students as a set.

Let the set of students playing basketball be “B” and that playing football be “F”. Then, students who do not play basketball is complement set B’ and students who do not play football is complement set F’. We have shown these complement sets separately for visualization. Actually, these complement sets are drawn to the same universal set, "U".

Two complement sets are but overlapping sets. There are students in the set B’ who play football and there are students in the set F’, who play basketball. In order to remove those students playing other game, we intersect two complements. The members of the intersection of two complements, therefore, represent students who play neither basketball nor football. This intersection is shown as third bottom Venn’s diagram in the figure.

Intersection

Intersection of two component sets

Looking at the intersection of two complement sets, however, we observe that this is equal to the complement of union “ B F ”. This conclusion can be derived from basic interpretation as well. We know that union “ B F ” represents students, who play either or both games. The component of the union, therefore, represents, who neither play basketball nor football.

This fact, as a matter of fact, is the first De-morgan’s law. Symbolically,

B F = B F

The second De-morgan’s law is :

B F = B F

In the parlance of illustration given earlier, let us interpret right hand side of the second De-morgan's law. The intersection “ B F ” represents students playing both games. Its complement, therefore, represents students who do not play both games, but may play one of them.

Component set

Component of intersection of two sets

Analytical proof

Here, we shall prove first De-morgan’s law in this section. The second law can be proved in similar fashion. Let us consider an arbitrary element “x” belonging to set ( A B )’.

x A B

x A B

Then, by definition of union,

x { x : x A o r x B }

Here, “not or” is interpreted same as “and”,

x A a n d x B

x A a n d x B

x A B

But, we had started with ( A B )’ 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 A B

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

A B A B

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

A B = A B

Example

Problem 1: In the reference of students in a class, the set “B” represents students, who play basketball. The set “F” represents students, who play football. The set “B” and “F” are left and right circles respectively on the Venn's diagram shown below. Identify regions marked 1 to 8 on the Venn’s diagram. Also interpret regions identified by combination U – (6+7).

Sets

Interpreting sets

Solution : The meaning of regions market 1 – 8 are as given hereunder :

1 : B-F : It represents the difference of “B” and “F”. It consists of students, who play basketball, but not football.

2 : F-B : It represents the difference of “F” and “B”. It consists of students, who play football, but not basketball.

3 : B F : It represents the intersection of two sets. It consists of students, who play both basketball and football.

4 : B: It represents the set “B”. It is union of two disjoint sets “B-F” and “ B F ”. It consists of students, who play basketball.

5 : F: It represents the set “F”. It is union of two disjoint sets “F-B” and “ B F ”. It consists of students, who play football.

6 : B∪F: It represents the union set of set “B” and “F”. Equivalently, it is union of three disjoint sets “B-F”, “ B F ” and “F-B”. It consists of students, who play either of two games or both.

7 : ( B F )’: It represents the component of union set “ B F ”. It consists of students, who play neither basketball nor football.

8 : B - F F - B : It represents union of two disjoint difference sets “B-F” and “F-B”. It consists of students, who play only one game.

The region, identified by U – (6+7), is complement of “ B F ”. It represents students, who do not play both games, but may play one of them.

Questions & Answers

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
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Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
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Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. 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
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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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