Question

If A and B are subsets of a set \[X\], then what is \[\left( A\cap \left( X-B \right) \right)\cup B\] is equal to?

Answer 1

Hint: In order to find the value of \[\left( A\cap \left( X-B \right) \right)\cup B\], we will be applying the properties of intersections and unions of sets. Intersection of sets means finding the common elements of two sets. In the same way, union of sets means considering all of the elements of both the sets.

Complete step-by-step solution:
Now let us have a brief regarding the set functions. A set function is a function whose domain is a collection of sets. Sets can be combined in a number of different ways to produce another set. Upon sets, we can perform operations such as union, intersection, subtraction, complement etc are performed in solving various set problems.
Now let us find the value of \[\left( A\cap \left( X-B \right) \right)\cup B\]
We will be finding the value by splitting the terms separately.
Firstly we will be considering \[\left( A\cap \left( X-B \right) \right)\], we can express this as
\[\left( A\cap \left( X-B \right) \right)=A-\left( A\cap B \right)\]
Since we have solved the first part, now we will be considering the complete function. We obtain as follows:
\[\left( A\cap \left( X-B \right) \right)\cup B=\left( A-\left( A\cap B \right) \right)\cup B\]
On solving this, we get
\[\Rightarrow \left( A\cup B \right)\]
\[\therefore \] \[\left( A\cap \left( X-B \right) \right)\cup B\]=\[\left( A\cup B \right)\].
Let us consider an example for the given set operation. Let \[X=\left\{ 1,2,3,4,5,6, \right\}\] and \[A=\left\{ 1,2,3 \right\}\] and \[B=\left\{ 5,6 \right\}\].
\[\left( A\cap \left( X-B \right) \right)\cup B\]=\[\left( A\cap \left( X-B \right) \right)\cup B=\left( \left\{ 1,2,3 \right\}\cap \left\{ 1,2,3,4 \right\} \right)\cup \left\{ 5,6 \right\}\]
\[\begin{align}
  & \Rightarrow \left( \left\{ 1,2,3 \right\}\cup \left\{ 5,6 \right\} \right) \\
 & =\left( 1,2,3,,5,6 \right)\Rightarrow \left( A\cup B \right) \\
\end{align}\]
Hence proved.

Note: The main error occurs when the braces are misplaced which leads to the obtaining of an inaccurate solution. So braces between the sets are to be correctly placed. We can also take the union of several sets simultaneously as well as the intersection can also be taken. The intersection as well as the union of the sets obeys the commutative as well as the associative property.