Question

Please prove this :-
(a minus b) union( B minus A) is equal to (a union B )minus( a intersection b).​

Answer 1

Answer:

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Answer 2

Answer:

This will be proved by taking an example.

Step-by-step explanation:

We will solve the left-hand side and right-hand side of the equation separately and by comparing the equation of both sides, by taking an example of two sets.

Let us take an example of two sets,

$\[\begin{gathered} A = \{ 2,4,6,8,10,12,14\} \hfill \\ B = \{ 1,3,5,7,9,11,13,15\} \hfill \\ \end{gathered} \]$

As we have to prove that:

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

Taking the RHS

$\[\begin{gathered} A - B = \{ 2,4,6,8,10,12,14\} \hfill \\ B - A = \{ 1,3,5,7,9,11,13,15\} \hfill \\ \end{gathered} \]$

Now,

$\[\left( {A - B} \right) \cup \left( {B - A} \right) = \{ 2,4,6,8,10,12,14,1,3,5,7,9,11,13,15\} \]............(1)$

Taking the LHS

$\[\begin{gathered} \left( {A \cup B} \right) = \{ 2,4,6,8,10,12,14,1,3,5,7,9,11,13,15\} \hfill \\ \left( {A \cap B} \right) = \left\{ \emptyset \right\} \hfill \\ \end{gathered} \]$

Now,

$\[\left( {A \cup B} \right) - \left( {A \cap B} \right) = \{ 2,4,6,8,10,12,14,1,3,5,7,9,11,13,15\} \]...............(2)$

By equations 1 and 2 we can say that,

LHS = RHS

Therefore,

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

Hence proved.