JEE Main & Advanced Mathematics Sets Operations on sets

(1) Union of sets : Let A and B be two sets. The union of A and B is the set of all elements which are in set A or in B. We denote the union of A and B by  \[A\cup B\], which is usually read as “A union B”.

Symbolically,         \[A\cup B=\{x:x\in A\,\,\text{or}\,\,x\in B\}.\]

(2) Intersection of sets : Let A and B be two sets. The intersection of A and B is the set of all those elements that belong to both A and B.

The intersection of A and B is denoted by \[A\cap B\] (read as “A intersection B”).

Thus, \[A\cap B=\{x\,\,:\,\,x\in \,\,and\,\,x\in B\}\].

(3) Disjoint sets : Two sets A and B are said to be disjoint, if \[A\cap B=\phi \]. If \[A\cap B\ne \phi \], then A and B are said to be non-intersecting or non-overlapping sets.

Example : Sets {1, 2}; {3, 4} are disjoint sets.

(4) Difference of sets : Let A and B be two sets. The difference of A and B written as \[A-B\], is the set of all those elements of A which do not belong to B.

Thus, \[A-B=\{x\,:\,x\in A\,\,and\,\,x\notin B\}\]

Similarly, the difference\[B-A\]  is the set of all those elements of B that do not belong to A i.e., \[B-A=\{x\in B:x\notin A\}\].

Example : Consider the sets \[A=\{1,\,2,\,3\}\] and \[B=\{3,\,4,\,5\}\], then \[A-B=\{1,\,2\};\,B-A=\{4,\,5\}\].

(5) 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\[A\Delta B\]. Thus, \[A\Delta B=\]\[(A-B)\cup (B-A)=\{x:x\notin A\cap B\}\].

(6) Complement of a set :  Let U be the universal set and let A be a set such that \[A\subset B\]. Then, the complement of A with respect to U is denoted by \[A'\] or \[{{A}^{o}}\] or \[C(A)\] or \[U-A\] and is defined the set of all those elements of U which are not in A.

Thus,       \[A'=\{x\,\in U\,:x\,\notin A\}\].

Clearly,    \[x\in A'\Leftrightarrow x\notin A\]

Example : Consider \[U=\{1,\,2,......,10\}\]

and \[A=\{1,\,3,\,5,\,7,\,9\}\].

Then \[{A}'=\{2,\,4,\,6,\,8,\,10\}\]