Set Operations

(i) Union:

If every elements of set  A and every elements of set  B are contained in a particular set, this set is called union of above sets  A and  B.

It is denoted by A\cup B

\therefore A\cup B=\{x |x \in A \text { or } x \in B \text { or } x \in \text { both } A \text { and } B\}

Venn Diagram:

Example:

 A = \{4,8,9,11\}

 B = \{2,3,5,7\}

 \begin{aligned} A\cup B&=\{4,8,9,11\}\cup \{2,3,5,7\}\\ &=\{2,3,4,5,7,8,9,11\}\end{aligned}

Note:

(a) A\cup \phi = A

(b) A\cup \bigcup = \bigcup

(c) A\cup A = A

 \rightarrow If  x \in A  \implies  x\in A \cup B and any x \in B  \implies x \in A \cup B

 \therefore A\subset A \cup B ,  B \subset A \cup B

(ii) Intersection:

The set of elements which belongs to both   A and  B, is called intersection of two sets  A and   B .

It is denoted by A \cap B .

 \therefore A \cap B= \{x|x\in A \text { and } x \in B\}

 \rightarrow If A \cap B = \phi , it means  A and  B are two disjoint sets.

Venn Diagram:

A \cap B

Example:

 A=\{2,3,4,6\} , B = \{1,2,3,4\}

A \cap B=\{2,3,4,6\} , \cap = \{1,2,3,4\}

=\{2,4\}

Note:

(a)  A \cap \phi =\phi

(b)  A \cap \bigcup= A

(c)  A \cap A = A

 \rightarrow If  x \in A \cap B \implies x \in A and any x \in A \cap B \implies x \in B

 \therefore  A \cap B \subset A , A \cap B \subset B

(iii) Complement of a set:

The set of all elements in the Universal set  \bigcup which do not belong to the set  A , is called complement of  A .

It is denoted by   A^c or  {A}'

 A^c=\bigcup-A = \{x| x \in \bigcup \text { and } x \notin A \}

i.e., for any x \in A ^c \Leftrightarrow x \notin A

(iv) Difference of sets:

The set of elements in  A which do not belong to  B , is called the difference from  A to  B .

It is denoted by  A-B, \rightarrow A-B =A\cap B^c

Example:

 A=\{2,3,4,7\} , B=\{7,8,9\}

A-B=\{2,3,4,7\} -\{7,8,9\} =\{2,3,4\}

B- A=\{7,8,9\}-\{2,3,4,7\} =\{8,9\}

Symmetric Difference of Sets:

The symmetric difference of two sets A and  B is denoted by A \Delta B .

It is defined as A \Delta B = (A-B) \cup (B-A)

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