Question

Show that \(A \oplus B = \left( {A \cup B} \right) - \left( {A \cap B} \right)\).

Short answer

Thus, we have proved that \(A \oplus B = \left( {A \cup B} \right) - \left( {A \cap B} \right)\).

Step-by-step solution

Definition

We know that the definition of symmetric difference,

i.e\(A \oplus B = \left( {A - B} \right) \cup \left( {B - A} \right)\).

Concept

here we have to prove,

\(\left( {A - B} \right) \cup \left( {B - A} \right) = \left( {A \cup B} \right) - \left( {A \cap B} \right)\)

Let \(x \in \left( {A \cup B} \right) - \left( {A \cap B} \right)\)

\( \Rightarrow x \in \left( {A \cup B} \right)\), \(x \notin \left( {A \cap B} \right)\)

\( \Rightarrow x \in A\) , \(x \in B\)but x is not in both sets

explanation

Case (1)

Suppose \(x \in A\) but \(x \notin {\bf{B}}\)

\( \Rightarrow x \in A - B\)

So,\(x \in \left( {A - B} \right) \cup \left( {B - A} \right)\)

Case (2)

Suppose

\(x \notin A\), \(x \in B\)

\( \Rightarrow x \in B - A\)

So, \(x \in \left( {A - B} \right) \cup \left( {B - A} \right)\)

Hence ,\(\left( {A \cup B} \right) - \left( {A \cap B} \right) \subseteq \left( {A - B} \right) \cup \left( {B - A} \right)\)._____________(i)

Converse part :- let \(x \in \left( {A - B} \right) \cup \left( {B - A} \right)\)

Either \(x \in A - B\) or \(x \in B - A\).

Case (1)

\( \Rightarrow x \in A - B\)then \(x \in A\)but \(x \notin B\)

So, \(x \in \left( {A \cup B} \right)\)but \(x \notin \left( {A \cap B} \right)\)

So \(x \in \left( {A \cup B} \right) - \left( {A \cap B} \right)\)

Case (2)

\(x \in B - A\)then but \(x \notin A\).

So, \(x \in \left( {B \cup A} \right)\) and \(x \notin \left( {B \cap A} \right)\)

So, \(x \in \left( {A \cup B} \right) - \left( {A \cap B} \right)\)

\(\left( {A - B} \right) \cup \left( {B - A} \right) \subseteq \left( {A \cup B} \right) - \left( {A \cap B} \right)\)___________(ii)

From (i) and (ii)

\(\left( {A - B} \right) \cup \left( {B - A} \right) = \left( {A \cup B} \right) - \left( {A \cap B} \right)\)

So, \(A \oplus B = \left( {A \cup B} \right) - \left( {A \cap B} \right)\).