Wednesday, 19 March 2014

Concept of Subset, Superset and Power Set

Consider the two sets A = {5, 4, 8} and B = {1, 5, 6, 4, 8}.
Do you notice any relation between sets A and B?
We can observe that every element of A is an element of B. In this case, we call A to be a subset of B. Mathematically, we write it as A  B.
If A and B are any two sets, then set A is said to be a subset of set B if every element of A is also an element of B. We write it as ⊆ B (read as ‘A is a subset of B’ or ‘A is contained inB’).
So, from the definition, we conclude that:
⊆ B if and only if ∈ A implies ∈ B.
Now, if ⊆ B, we can also say that B contains A. In this case, we say that B is a superset of A. We write it as B ⊇ A (read as ‘B contains A’ or ‘B is a superset of A’).
Now, consider the two sets A = {letters of FOLLOW} and B = {letters of LOWER}.
Then, A = {F, O, L, W} and B = {L, O, W, E, R}
Is set A a subset of set B?
Observe that there is an element, F in set A which is not a member of set B.
So, A is not a subset of B. We write ‘A is not a subset of B’ as ‘ B’. This is also read as ‘A is not contained in B’.
If there exists at least one element in A which is not an element of B, then A is not a subset of B. Mathematically, we write it as A  B
Another concept you need to know about is that of proper subsets.
To understand what we mean by a proper subset, let us look at the two sets given below.
A = {4, 8, 12}
B = {2, 4, 6, 8, 10, 12, 14}
Observe that A is a subset of BIs there any element in B which is not a member of A?
Yes, 2, 6, 10, 14 ∈ B; however, 2, 6, 10, 14  A.
So, in this case, we say that A is a proper subset of B.
Let A be any set and B be a non-empty set. Set A is called a proper subset of B if and only if every member of A is also a member of B, and there exists at least one element in B which is not a member of A. We write it as ⊂ B.
In simple language, we say that A is a proper subset of B, i.e., A ⊂ B, if A is a subset of B and A ≠ B.
Conversely, if two sets A and B are such that A  B and A ≠ B, then A is called a proper subset of Band B is called  the superset of A
The following are some important points to be noted.
(a) Every set is a subset of itself.
(b) A subset which is not a proper subset is called an improper subset. If A and B are two equal sets, then A and B are improper subsets of each other.
(c) Every set has only one improper subset and that is itself.
(d) An empty set is a subset of every set.
(e) An empty set is a proper subset of every set except itself.
(f) Every set is a subset of the universal set.
(g) If  Y and ⊆ X, then X = Y
Now, when given any finite set A, we know how to find its cardinal number. Can we also find the number of subsets and the number of proper subsets of set A?
Yes, we can.
To know what these numbers are, let us suppose that the cardinal number of the set A is m, i.e., (A) =m, then
The number of subsets of A = 2m
The number of proper subsets of A = 2 1
For example:
If A = {1, 3, 5}, then (A) = 3.
∴ Number of subsets of A = 23 = 8
These are: Φ, {1}, {3}, {5}, {1, 3}, {1, 5}, {3, 5}, {1, 3, 5}
Number of proper subsets of A = 2 1 = 8  1 = 7
Now, let us discuss about power set.
The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.
If the number of elements in set A is m, then the number of elements in the power set of A is 2m.
i.e., nP(A) = 2m, where n(A) = m
We have another type of a set called universal set and it can be defined as follows:
A set that contains all the elements under consideration in a given problem is called universal set and it is denoted by U or S or ξ.
Let us consider three sets XY, and U as:
X = {the consonants of English alphabets}
Y= {all the letters of the word EDUCATION}
= {all the letters in English alphabet}
Did you notice any relation between the sets XY, and U?
Let us see.
We have, X = {B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, W, X, Y, Z}, Y = {E, D, U, C, A, T, I, O, N} and U = {A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z}
Observe that the set U contains all the elements of both the sets X and Y. In this, we say that set U is auniversal set for the sets X and Y
Universal set may vary from problem to problem.
For example, if we consider a set as {0, 1}, then we may consider its universal set as W or Z or {–1, 0, 1, 2} etc. Similarly, if we consider the set as {Ganga, Yamuna, Saraswati}, then we may consider the universal set as {the rivers in India}.
Therefore, we should always specify the universal set for a given problem.
Now, let us learn to represent the information related to above discussed concepts using Venn diagrams.
Look at the following diagram. 
Here, UAB and C are four sets. 
It can be seen that A is the subset of B i.e., A ⊆ B.
Also, set A is completely contained by set B i.e., set ⊂ set A, we can say that A?B.
Similarly, set C is neither contained by set B nor by set A. Thus, C  and C  A.
Also, it can be seen that set U contains all of the sets AB and C. Thus, U is universal set.
Let’s now look at some examples to improve our understanding of the above discussed concepts.
Example 1:
With respect to the three sets: A = {5, 10, 15, 20}, B = {1, 2, 3, …, 20}, and C = {2, 4, 6, 8, 10, 12, 16, 18, 20}, classify the following statements as true or false?
(a) A⊂B
(b) B⊆ C
(c) A  C
(d) Φ⊆A
(e) B⊆ U
(a) True
Since 5, 10, 15, 20  A and 5, 10, 15, 20  B
Also, there exist many elements which are a member of set B, but not of set A.
So, A⊂ B
(b) False
Since 1, 3, 5, 7, 9, 11, 13, 15, 17, 19  B and these elements do not belong to C
So,  C
(c) False
Since 5, 15  A, but 5, 15  C
So, A  C
(d) True
Since an empty set is a subset of every set
So, Φ  A
(e) True.
Since every set is a subset of the universal set
So, B  U
Example 2:
Write all the subsets of the set {3, 6, 9, 12}. Which of these are proper subsets and which are improper subsets?
Let A = {3, 6, 9, 12}
The subsets of set A are:
The proper subsets are
The improper subset is {3, 6, 9, 12}, i.e., set A itself.
Example 3: 
Three sets are defined as A = {1, 3, 4}, B = {3, 4} and C = {3, 4, 2, 1}. Prove that A and C are the super sets of B.
We have,
A = {1, 3, 4}, 
B = {3, 4} and
C = {3, 4, 2, 1}
It can be seen that all the elements of B are also in sets A and C. Also, B ⊂ A and B ⊂ C. This means that B is a proper subset of A as well as C.
 B  A and B ⊂ C.
Thus, A and C are the supersets of B.
Example 4:
If A = {1, 2, 3, 4, 5}, then find the number of subsets of set A. Also, find the number of proper subsets of set A.
A = {–1, –2, –3, –4, –5},
∴ n (A) = 5 = m (say)
Number of subsets of A = 2m = 25 = 32
Number of proper subsets of A = 2m – 1 = 32 – 1 = 31
Example 5:
Write the power set for the set A = {1, 3, 5}.
First, let us write all the subsets of set A.
{Φ}, {1}, {3}, {5}, {1,3}, {1,5}, {3,5}, {1,3,5}
P(A) = {{Φ}, {1}, {3}, {5}, {1,3}, {1,5}, {3,5}, {1,3,5}}
Example 6:
Given set = {xx is a natural number less than 10} and set B = {yy is an even number less than 9}. Is set B a subset of set A? If yes then draw the Venn-diagram depicting the given sets.
Given, A = {xx is a natural number less than 10} 
i.e., A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
B = {yy is an even number less than 9}
i.e., B = {2, 4, 6, 8}
It is clear that B ⊂ A.
Venn-diagram for B ⊂ A is shown below:
Example 7:
Let A and B be two finite sets such that n(A) = m and n(B) = n. If the ratio of the number of elements of power sets of A and B is 64 and n(A) + n(B) = 32, find the value of m and n.
Given that
n(A) = m
n(B) = n
The ratio of the number of elements of power sets of A and B is 64.
∴ m – n = 6    ...(1)
Also, n(A) + (B)= 32
∴ m + n = 32    ...(2)
From equations (1) and (2), we get
m 19
n 13
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