Question: If \[A=\left\\{ 1,2,3 \right\\}\], \[B=\left\\{ 3,4 \right\\}\] and \[C=\left\\{ 4,5,6 \right\\}\], ...

Question

If $A=\left\\{ 1,2,3 \right\\}$ , $B=\left\\{ 3,4 \right\\}$ and $C=\left\\{ 4,5,6 \right\\}$ , then $A\cup \left( B\cap C \right)$ is equal to

Answer 1

Solution

In order to find $A\cup \left( B\cap C \right)$ , firstly we will be finding the set of $\left( B\cap C \right)$ which means we will only be considering the common element from the sets B and C. Then after finding the set of $\left( B\cap C \right)$ , we will be combining the complete set of A and express them together as union set operation takes place. This would be our required answer.

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, operations such as union, intersection, subtraction, complement etc are performed in solving various set problems.
Now let us start solving our problem.
So we are given three sets. They are:
$A=\left\\{ 1,2,3 \right\\}$ ,
$B=\left\\{ 3,4 \right\\}$ ,
$C=\left\\{ 4,5,6 \right\\}$
Firstly, let us find $\left( B\cap C \right)$ which means considering only the common elements in sets B and C. We get that it is 4.
$\left( B\cap C \right)=\left\\{ 4 \right\\}$
Now let us consider $A\cup \left( B\cap C \right)$ .
So, we have to take elements present in both the sets, so we get that elements are 1, 2, 3 and 4.
$A\cup \left( B\cap C \right)=\left\\{ 1,2,3,4 \right\\}$

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.