Sets

Cardinal number of set:  
The number of distinct elements contained in a finite set is called its cardinal number and is denoted by n(A)
Example:  If  A={1,2,3,4,5} then n(A)=5,  if    B={1,2,3} n(B)=3
Subsets:  Let A, B be two sets such that every member of A is a member of B, then A is called a subset of B, it is written as \(A\subseteq B\) .
Thus, \(A \subseteq B\)  iff (read as ‘if and only if’) \(x \in A \implies x\in B\)
    If \(\exists \) (read as ‘there exists) at least one element in A which is not a member of B, then A is not a subset of B and we write it as \(A \not\subset B\)
For Example:
    i)    Let A={-1,2,5}  and  B={3,-1,2,7,5} the \(A\subset B\) . Note that  \(B \not\subset A\).
    ii) The set of all even natural numbers is a subset of the of natural numbers.
Some properties of subsets:
    i)    The null set is subset of every set. Let A be any set.
     \(\varnothing \subseteq A,\) as there is no element in \(\varnothing \) which is not in A.
    ii) Every set is subset of itself. Let A be any set.
        \(x \in A \implies x \in A \therefore A \subseteq A\)               
    iii) If \(A\subseteq B\) and \(B \subseteq C\) then \(A \subseteq C\) Let \(x\in A\)
       \(x\in B\)         \((\because A \subseteq B)\)     \(\because x\in C\)   \((\because B \subseteq C)\)     \(\because A \subseteq C\)     .
    iv) A=B iff  \(A\subseteq B~~and~B\subseteq A\) . Let A =B.
          \(\begin{align} & \therefore x\in A\Rightarrow x\in B\left( \because A=B \right) \\ & A\subseteq B\text{similarly,}x\in B\Rightarrow x\in A\left( \because A=B \right) \\ & B\subseteq A\text{ }conversely\text{,}~\text{let }A\subseteq B~~and~B\subseteq A \\ & \therefore x\in A\Rightarrow x\in B\left( \because A\subseteq B \right),and~x\in B\Rightarrow x\in A \\ \end{align} \)  
  it follows that every set is a subset of itself. 
Notes: 
1.     Two sets A and B are equal iff  \(A\subseteq B~~and~B\subseteq A\)
2.      Since every element of a set A belongs to A, it follows that every set is a subset of itself.
        Proper subset: Let A be a subset of B. We say that A is proper subset of B if  i.e., i there exists at least one element in B which does not belong to A. A subset, which is not proper, is called an improper subset.
Example: 1. If  A={1,2,3}  then proper subsets of A are \(\varnothing ,\), {1},{2},{3},{1,2},{1,3},{2,3}
Example: 2. A= {1,2,3,4,5} ,B={2,3,4}
    Every element of B i.e., 2, 3 and 4 is also an element of A.
    \(\therefore B \subset A\)
    Further we note that there are two more elements that are in A and not in B. They are 1 and 5. Then in such circumstances we say that B is a proper subset of A. 
Example: 3 \(N\subset W\subset Z\subset Q\subset R\)
Remark: 1. If \(A \subseteq B\) then every element of A is in B and there is a chance that A may be equal to B i.e., every element of B is A, but if , then every element of A is in B and there is no chance that A may be equal to B i.e., there will exist at least one element in B which is not in A.
     \(\begin{align} & A\subset B\Rightarrow A\subseteq B \\ & i.e.,A\subseteq B,B\not\subset A \\ \end{align}\) 
Remark 2: If \(A \subseteq B\) we may have \(B\subseteq A\) but if \(A \subset B\), we cannot have \(B\subset A\)
Power set: The set formed by all the subsets of a given set A is called the power set of A it is usually denoted by P(A).
For example: i) Let A={0}, then p(A)={\(\varnothing \) ,{0}}. Note that N(P(A))=2=\(2^1\).
    Note that O(P-(A))=8=\(2^3\) in all these examples, we have observed that N(P(A))=\(2^{n(A)}\)
    Rule to write down the power set of a finite set A:
    First of all write \(\varnothing \) .
    Next, write down singleton subsets each containing only one element of A.
    In the next step write all the subsets which contain two elements from the set A.
    Continue this way and in the end write A itself as A is also a subset of A.
    Enclose all these subsets in braces to get the power set of A.
    Comparable sets: Two sets A and B are said to be comparable iff either \(A\subset B\)  or \(B\subset A\)
For example: i) The sets A={1,2} and B={1,2,4,5} are comparable as \(A \subset B\).
Universal set: In any application of the theory of sets, all sets under investigation are regarded as subsets of fixed set. We call this set the universal set, it is usually denoted by X or U