Question

Show that for any sets A and B ,
\[A = \left( {A \cap B} \right) \cup \left( {A - B} \right){\text{ and }}A \cup \left( {B - A} \right) = \left( {A \cup B} \right).\]

Answer 1

Hint : In this question we will use the operations on sets to prove this statement. By taking some examples of sets and then by checking left hand side and right hand side statements we will solve this problem. We can also solve this question with the help of venn diagrams easily.

Complete step-by-step answer:
Here ,let us take two sets A and B for example ,
$
   \Rightarrow U = \{ 1,2,3,4,5,6,7,8\} \\
   \Rightarrow A = \{ 1,2,3,4,5\} \\
   \Rightarrow B = \{ 4,5,6,7,8\} \\
 $
Now , use these sets and apply some sets operations on the elements of these sets .
1. for\[A = \left( {A \cap B} \right) \cup \left( {A - B} \right)\]. …….(i)
We have , $A = \{ 1,2,3,4,5\} $and $B = \{ 4,5,6,7,8\} $
$
   \Rightarrow \left( {A \cap B} \right) = \{ 1,2,3,4,5\} \cap \{ 4,5,6,7,8\} \\
  {\text{ }}\left( {A \cap B} \right) = \{ 4,5\} \\
 $ ………(ii)
$
   \Rightarrow \left( {A - B} \right) = \{ 1,2,3,4,5\} - \{ 4,5,6,7,8\} \\
  {\text{ }}\left( {A - B} \right) = \{ 1,2,3\} \\
$ ………(iii)
Now , taking union of equation (ii) and (iii), we get
$
   \Rightarrow \left( {A \cap B} \right) \cup \left( {A - B} \right) \\
   \Rightarrow\{ 4,5\} \cup \{ 1,2,3\} \\
   \Rightarrow\{ 1,2,3,4,5\} \\
$
Now , by comparing this with set A , we get
$
  A = \left( {A \cap B} \right) \cup \left( {A - B} \right) \\
  \{ 1,2,3,4,5\} = \{ 1,2,3,4,5\} \\
$
Hence this statement is true .
2. for\[A \cup \left( {B - A} \right) = \left( {A \cup B} \right)\]
Now ,
$
   \Rightarrow \left( {A \cup B} \right) = \{ 1,2,3,4,5\} \cup \{ 4,5,6,7,8\} \\
   \Rightarrow \left( {A \cup B} \right) = \{ 1,2,3,4,5,6,7,8\} \\
 $ …….(iv)
$
   \Rightarrow \left( {B - A} \right) = \{ 4,5,6,7,8\} - \{ 1,2,3,4,5\} \\
   \Rightarrow \left( {B - A} \right) = \{ 6,7,8\} \\
$
Now, \[A \cup \left( {B - A} \right)\]
$
   \Rightarrow A \cup \left( {B - A} \right) = \{ 1,2,3,4,5\} \cup \{ 6,7,8\} \\
   \Rightarrow A \cup \left( {B - A} \right) = \{ 1,2,3,4,5,6,7,8\} \\
$……..(v)
From equation (iv) and (v), we get
$
   \Rightarrow A \cup \left( {B - A} \right) = \left( {A \cup B} \right) \\
   \Rightarrow \{ 1,2,3,4,5,6,7,8\} = \{ 1,2,3,4,5,6,7,8\} \\
$
 PROVED
Hence ,we have proved both the statements by using set operation .
Note : In this question ,we have used some basic set operations on the sets that we have taken for example to prove the statements given in the question ,then we make the sets that are needed . After that we compared the left hand side and right hand side so that we can identify that the statement is true .