Subsets:- Consider a set ‘A’ of all the boys in class
eleventh and set ‘B’ of all the students ( Boys + Girls) in class eleventh. It
is clear that all the members of set ‘A’ are also members of set ‘B’ then we
can say that set ‘A’ is subset of set ‘B’ or set ‘A’ is contained in set ‘B’
and write it as
A⊂B (read as A is subset
of B or A is contained in B)
or B⊃A (read as Bis superset of A or B contains A)
We can also express it as
A⊂B if a∈A ⇒ a∈B
If all the members of any set say C are
not in any other set say D then we say that C is not a subset of set D and
write it as
C⊄D (read it as C is not a subset of D or C is
not contained in D)
Subset of a set may be Proper Subset
or Improper Subset.
Proper Subset :- If
a subset does not contain all the elements of its superset then it is called
proper subset but if it contains all the elements of its superset then it is
called improper subset.
Let
us understand it with the help of an example
Take C = {1,2,3}, D = {1,2,3,4,5} and E = {1,2,3,4,5}
Here C is proper subset of E, we write it as C⊂E
and D is improper subset of E, we write it as D⊆E
An empty set is subset of every set.
Number of subsets of a given set:-
Let us take sets F,G,H & C such that
F = φ ( number of elements = 0)
Its subset is φ (Number of subsets = 1 = 20)
G = {1} ( number of elements = 1)
Its subsets are φ and {1} (Number of subsets = 2 = 21)
H = {1,2} ( number of elements = 2)
Its subsets are φ, {1}, {2} and {1,2} (Number of subsets = 4 = 22)
C = {1,2,3} ( number of elements = 3)
Its subsets are φ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} (Number of subsets = 8 = 23)
From above results it is clear that if a set contains ‘n’ elements then number of all of its subsets is 2n (Including proper and improper subsets)
i.e. Total number of subsets = 2n
Total number of improper subsets = 1
Total number of proper subsets = 2n -1
Comparable sets:- Two sets are comparable if either of them is subset of remaining set
i.e. if either A⊆B or B⊆A
Universal set:- If we are discussing about any number of sets and each set is subset of a particular set (say ’U’) then U is called Universal set of sets under discussion.
Power set:- It is the set of all the subsets of a given set.
For set H = {1,2}
Its subsets are φ, {1}, {2} and {1,2}
Then power set of H [denoted by P(H)] is
P(H) = { φ, {1}, {2}, {1,2}}