Question

If $A = \\{ 1,2,3,4\\}$,$B = \\{ 3,4,5,6\\}$, $C = \\{ 5,6,7,8\\}$, $D = \\{ 7,8,9,10\\}$, find:
(1) $A \cup B$
(2) $A \cup C$
(3) $B \cup C$
(4) $B \cup D$
(5) $A \cup B \cup C$
(6) $A \cup B \cup D$
(7) $B \cup C \cup D$

Answer 1

Here we must know what the symbol means. This symbol represents the union of the sets which are there. For example: If we are given $A \cup B$ then it represents the union of the two sets $A{\text{ and }}B$ which means that in $A \cup B$ there will be all the elements which are present in the sets $A$ and $B$ where the elements which are common in both the sets need to be counted only once not twice.

Complete step-by-step answer:
Here we are given the four sets which are sets $A,B,C,D$ and there are the elements which are present in the individual sets as
$A = \\{ 1,2,3,4\\}$,$B = \\{ 3,4,5,6\\}$, $C = \\{ 5,6,7,8\\}$, $D = \\{ 7,8,9,10\\}$
Here in this question we need to solve the several parts where the symbol is present. So we must know what this symbol means. This symbol represents the union of the sets which are there. For example: If we are given $A \cup B$ then it represents the union of the two sets $A{\text{ and }}B$ which means that in $A \cup B$ there will be all the elements which are present in the sets $A$ and $B$ where the elements which are common in both the sets need to be counted only once not twice.
Solving the given parts:
(1) $A \cup B$
Here we know that$A = \\{ 1,2,3,4\\}$,$B = \\{ 3,4,5,6\\}$
So we need to find the common elements of both the sets whose union is to be found.
So we get that there are two elements common in both the sets which are $\\{ 3,4\\}$
So we need to put these two elements only once in the $A \cup B$
So $A \cup B$ $= \\{ 1,2,3,4,5,6\\}$
(2) $A \cup C$
Here we know that$A = \\{ 1,2,3,4\\}$,$C = \\{ 5,6,7,8\\}$
So we need to find the common elements of both the sets whose union is to be found.
So we get that there is no element which is common in both the sets so union of these two sets will simply be the representation of all the elements in the $A \cup C$
So we write all the elements of set A and C in $A \cup C$
So $A \cup C$ $= \\{ 1,2,3,4,5,6,7,8\\}$
(3) $B \cup C$
Here we know that$B = \\{ 3,4,5,6\\}$, $C = \\{ 5,6,7,8\\}$
So we need to find the common elements of both the sets whose union is to be found.
So we get that there are two elements common in both the sets which are $\\{ 5,6\\}$
So we need to put these two elements only once in the $B \cup C$
So $B \cup C$ $= \\{ 3,4,5,6,7,8\\}$
(4) $B \cup D$
Here we know that $B = \\{ 3,4,5,6\\}$,$D = \\{ 7,8,9,10\\}$
So we need to find the common elements of both the sets whose union is to be found.
So we get that there is no element which is common in both the sets so union of these two sets will simply be the representation of all the elements in the $B \cup D$
So we write all the elements of set B and D in $B \cup D$
So $B \cup D$ $= \\{ 3,4,5,6,7,8,9,10\\}$
(5) $A \cup B \cup C$
Here we need to find the union of the three sets $A,B,C$
We know that
$A = \\{ 1,2,3,4\\}$,$B = \\{ 3,4,5,6\\}$, $C = \\{ 5,6,7,8\\}$
Hence we need to see that the common elements are to be written only once. So we can write that
$A \cup B \cup C$ $= \\{ 1,2,3,4,5,6,7,8\\}$
(6) $A \cup B \cup D$
Here we need to find the union of the three sets $A,B,D$
We know that
$A = \\{ 1,2,3,4\\}$,$B = \\{ 3,4,5,6\\}$, $D = \\{ 7,8,9,10\\}$
Hence we need to see that the common elements are to be written only once. So we can write that
$A \cup B \cup D$ $= \\{ 1,2,3,4,5,6,7,8,9,10\\}$
(7) $B \cup C \cup D$
Here we need to find the union of the three sets $B,C,D$
We know that
$B = \\{ 3,4,5,6\\}$,$C = \\{ 5,6,7,8\\}$, $D = \\{ 7,8,9,10\\}$
Hence we need to see that the common elements are to be written only once. So we can write that
$B \cup C \cup D$ $= \\{ 3,4,5,6,7,8,9,10\\}$

Note: To solve this type of question where we are given the symbol relating the two sets, we must have the proper knowledge of what the symbol represents otherwise our whole question will go wrong.
For example: $A \cup B$ represents the union of the two sets A and B
The $A \cap B$ means the intersection of the two sets which means the common elements of A and B
The $A \subset B$ means the A is subset of B
Hence in this way we must have the complete knowledge of the symbols’ representation.
One main point for this question is that we need to write the common elements only once in union of the elements contained in the sets.