Before studying about the Complement of a set, let us understand what are sets?
Sets Definition
A well-defined collection of objects or elements is known as a set. Any set consisting of all the objects or elements related to a particular context is defined as a universal set. It is represented by \(\begin{array}{l}U\end{array} \)
.
For any set A which is a subset of the universal set \(\begin{array}{l}U\end{array} \)
, the complement of the set \(\begin{array}{l}A\end{array} \)
consists of those elements which are the members or elements of the universal set \(\begin{array}{l}U\end{array} \)
but not of the set \(\begin{array}{l}A\end{array} \)
. The complement of any set \(\begin{array}{l}A\end{array} \)
is denoted by \(\begin{array}{l}A'\end{array} \)
.
Also, read:
Complement of a Set Definition
If \(\begin{array}{l}U\end{array} \)
is a universal set and \(\begin{array}{l}A\end{array} \)
be any subset of \(\begin{array}{l}U\end{array} \)
then the complement of \(\begin{array}{l}A\end{array} \)
is the set of all members of the universal set \(\begin{array}{l}U\end{array} \)
which are not the elements of \(\begin{array}{l} A\end{array} \)
.
\(\begin{array}{l}A'\end{array} \)
= {\(\begin{array}{l}{x ~:~ x~ ∈ ~U ~and ~x ~∉ ~A}\end{array} \)
}
Alternatively it can be said that the difference of the universal set \(\begin{array}{l}U\end{array} \)
and the subset \(\begin{array}{l}A\end{array} \)
gives us the complement of set \(\begin{array}{l}A\end{array} \)
.
Venn Diagram for the Complement of a set
The Venn diagram to represent the complement of a set A is given by:
Complement of a Set Examples
To make it more clear consider a universal set \(\begin{array}{l}U\end{array} \)
of all natural numbers less than or equal to 20.
Let the set \(\begin{array}{l}A\end{array} \)
which is a subset of \(\begin{array}{l}U\end{array} \)
be defined as the set which consists of all the prime numbers.
Thus we can see that \(\begin{array}{l}A\end{array} \)
= {\(\begin{array}{l}{2, 3, 5, 7, 11, 13, 17, 19}\end{array} \)
}
Now the complement of this set A consists of all those elements which is present in the universal set but not in \(\begin{array}{l}A\end{array} \)
. Therefore, \(\begin{array}{l}A'\end{array} \)
is given by:
\(\begin{array}{l}A'\end{array} \)
={\(\begin{array}{l}{1,4,6,8,9,10,12,14,15,16,18,20}\end{array} \)
}
Example: Let \(\begin{array}{l}U\end{array} \)
be the universal set which consists of all the integers greater than 5 but less than or equal to 25. Let \(\begin{array}{l}A\end{array} \)
and \(\begin{array}{l}B\end{array} \)
be the subsets of \(\begin{array}{l}U\end{array} \)
defined as:
\(\begin{array}{l}A\end{array} \)
= {\(\begin{array}{l}{x~:x~ ∈~U ~and~ x~ is~ a~ perfect~ square}\end{array} \)
}
\(\begin{array}{l}B\end{array} \)
= \(\begin{array}{l}{7, 9, 16, 18, 24}\end{array} \)
Find the complement of sets A and B and the intersection of both the complemented sets.
Solution: The universal set is defined as:
\(\begin{array}{l}U\end{array} \)
= {\(\begin{array}{l}{6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}\end{array} \)
}
Also, \(\begin{array}{l}A\end{array} \)
= {\(\begin{array}{l}{9,16,25}\end{array} \)
} and
\(\begin{array}{l}B\end{array} \)
= {\(\begin{array}{l}{7,9,16,18,24}\end{array} \)
}
The complement of set A is defined as:
\(\begin{array}{l}A'\end{array} \)
= {\(\begin{array}{l}{x~:~x~∈~U~ and ~x~∉~A}\end{array} \)
}
Therefore, \(\begin{array}{l}A'\end{array} \)
= {\(\begin{array}{l}{6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,23,24}\end{array} \)
}
Similarly the complement of set B can be given by:
\(\begin{array}{l}B'\end{array} \)
= {\(\begin{array}{l}{6,8,10,11,12,13,14,15,17,19,20,21,22,23,25}\end{array} \)
}
The intersection of both the complemented sets is given by \(\begin{array}{l}A’∩ B'\end{array} \)
.
Rightarrow A’∩ B’= {6, 8, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23}
We can see from the above discussions that if a set \(\begin{array}{l}A\end{array} \)
is a subset of the universal set \(\begin{array}{l}U\end{array} \)
then the complement of set \(\begin{array}{l}A\end{array} \)
i.e. \(\begin{array}{l}A'\end{array} \)
is also a subset of \(\begin{array}{l}U\end{array} \)
..
Now we are clear on the concept of the complement of sets. There is a lot more to learn. Enrich your knowledge and reach new horizons of success by downloading BYJU’S – The Learning App to know more or visit our website.
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