Math

Universal Set :
All sets related to the discussions are subset of a definite set. Such as, A 􀀠 {x, y} is a subset of
B 􀀠 {x, y, z}. Here, set B is called the universal set in with respect to the set A.
So, if all the sets under discussion are subsets of a particular set, that particular set is
called the universal set with respect to its subsets.
Universal set is generally expressed by U . But universal set can
be expressed by other symbols too. Such as, if set of all even
natural numbers C = {2, 4, 6........} and set of all natural
numbers N= {1, 2, 3, 4,...........}, the universal set with respect
to C will be N .

Complement of a Set :
U is an universal set and A is the subset of U. The set formed
by all the elements excluding the elements of set A is called the
complement of set A . The complement of the set A is
expressed by c A or A􀁣 . Mathematically, A U A c 􀀠 \
Let, P and Q are two sets and the elements of Q which are not
elements of P are called complement set of Q with respect to P and written as
Q P \ Q.

Example 5. Write the subsets of P ={x, y, z} and find the proper subset from the
subsets.
Solution : iGven, P ={x, y, z}
Subsets of P are {x, y, z} , {x, y}, {x, z} , {y, z}, {x} , {y}, {z}, φ .
rPoper subsets of P are {x, y}, {x, z} , {y, z}, {x} , {y}, {z}
Equivalent Set :
If the elements of two or more sets are the same, they are called equivalent sets. Such
as, A ={3, 5, 7} and B = {5, 3, 7} are two equal sets and written as A =B .
Again, if A = {3, 5, 7}, B = {5, 3, 3, 7} and C ={7, 7, 3, 5, 5} , the sets A, B and C
are equivalent. That is, A =B =C