Sets

OPERATIONS OF SETS
UNION OF SETS
    The union of two sets A and B is the set of all those elements, which are either in A or in B ( including those which are in both)
    In symbolic form, union of two sets A and B is denoted as, . It is read as “A union B”.
    \(\begin{align}   & \therefore x\in \left( A\cup B \right)\Rightarrow x\in A(or)x\in B \\  & And,x\notin \left( A\cup B \right)\Rightarrow x\notin A(or)x\notin B \\ \end{align}\)
    It is evident from definition that  \(A\subseteq \left( A\cup B \right);B\subseteq \left( A\cup B \right)\)
SOLVED EXAMPLES    
    i)  A={a,e,i,o,u},b={a,b,c}
    ii) A={1,3,5},B={1,2,3}
Solution:- i) We have, \(A\cup B=\{a,e,i,o,u\}\cup \{a,b,c\}\Rightarrow A\cup B=\{a,b,c,e,i,o,u\}\)
    Have, the common element a has been taken only once while writing .
UNION OF THREE OR MORE SETS
    The union of \(n(n\ge 3)\) for sets \(A_1,A_2,A_3,.......A_n\) is defined as the set of all those elements which are in \(A_1(1\le i \le n)\) for at least one value of i. The union of \(A_1,A_2,A_3,.......A_n\) is denoted \(\bigcup\limits_{i=1}^{n}{{{A}_{1}}}\) In symbols, we write  \(\bigcup\limits_{i=1}^{n}{{{A}_{1}}}=\{x:x\in {{A}_{1}}\text{For atleast one valueof i},1\le i\le n\}\)
INTERSECTION OF SETS: 
    The intersection of two sets A and B is the set of all those elements which belong to both A and B.
    Symbolically, we write \(A\cap B=\{x:x\in Aandx\in B\}\) and read as “ A intersection B” 
    Let \(x\in(A\cap B)\implies x\in A\) and \(x\in B\) and  \(x\notin \left( A\cap B \right) \implies x\notin A ( or )x\notin B\) 
    It is evident from the definition that  \(A\cap B\subseteq A,A\cap B\subseteq B\)
SOLVED EXAMPLES:
EXAMPLES: 1.i) If  A={1,3,5,7,9,11} ,B={7,9,11,13}  find \(A\cap B\)
                 ii) If A={a,b,c} , B=\(\varnothing \) find  
Solution.    i) Since, 7, 9, 11 are the only elements which are common to both the sets A and B.
    ii)Since, there is no common element.

INTERSECTION OF MORE SETS
    The intersection of   sets  A₁, A₂, A₃ .........A_n is defined as the set of all the elements which are in \(A_i(1\le i \le n)\) for each i.
    The intersection of  A₁, A₂, A₃ .........A_n is denoted by \(\bigcup\limits_{i=1}^{n}{{{A}_{1}}}\)
    In symbols, we write,\(\bigcap\limits_{i=1}^{n}{{{A}_{1}}}=\{x:x\in {{A}_{1}}\text{For all i},1\le i\le n\}\)
DIS JOINT SETS: Two sets A and B are said to be disjoint, if A B = . If A B, then
    A and B are said to be intersecting sets or overlapping sets. E.g. Let A = {1, 2, 3},B ={a, b, c}AB=,hence A and B are disjoint.
Difference of sets : If A and B are two sets, then their difference A-B is the set of all those elements of A which do not belong to B
    Let XA-BXA and XB. Similarly, the difference B-A is the set of all those elements of B that do not belong to A. i. e.,
     \( B - A = \left\{ {x:x \in B\,and\,x \in A} \right\}\)
     \( x \in B - A \Leftrightarrow x \in B\,\,and\,x \notin A\)
Symmetric Difference of two sets: 
    Let A and B be two sets .The symmetric difference of sets A and B is the set   
     (A-B)\( \cup \)(B-A) and is denoted by AB
     \( \therefore A\Delta B = (A - B) \cup (B - A)\,or\,\{ A \cup B\} - \{ A \cap B\} \)
For example : Let A = {1, 3, 5, 7, 9},B = {2, 3, 5, 7, 11}
    Then, A-B = {1, 9}; B-A = {2, 11}
     A\( \Delta \)B = {A-B} \(\cup\) {B-A} = {1, 9}\(\cup\){2, 11} = {1, 2, 9, 11}
Compliment of set : Let U be the universal set and A is a subset of U. Then ,the compliment of A with respect to (w. r. t) U is the set of all elements of U which are not the elements of A. Compliment of A with respect to U is denoted by A¹  or A^c
    In symbolic from \( {{\text{A}}^1}{\text{ = \{ x : x}} \in {\text{ U and x }} \notin {\text{ A\} clearly, }}{{\text{A}}^1}{\text{ = U - A}}\)  
    For example: If U = {1, 2, 3, 4, 5, 6} and A ={2, 4, 6}
    Then, A¹ =U – A = {1, 2, 3, 4, 5, 6} – {2, 4, 6}  A^₁  = {1, 3, 5}
Some results on complementation: 
\( \begin{gathered} {\text{1}}{\text{.}} & {U^|}{\text{ = \{ x }} \in \phi {\text{ :X}} \notin {\text{ U\} = }}\phi \hfill \\ {\text{2}}{\text{.}} & {\phi ^|}{\text{ = \{ X}} \in {\text{ U : x}} \in \phi {\text{\} = U}} \hfill \\ {\text{3}}{\text{.}} & {({A^|}{\text{)}}^|}{\text{ = \{ x}} \in {\text{ U : x}} \notin {{\text{A}}^|}{\text{ \} = \{ x}} \in {\text{U : x}} \in A{\text{\} = A}} \hfill \\ {\text{4}}{\text{.}} & {\text{A}} \cup {{\text{A}}^|}{\text{ = \{ x}} \in {\text{U : x}} \in {\text{A\} \{ x}} \in U{\text{ : x }} \notin {\text{A\} = U}} \hfill \\ \end{gathered} \)