Question

If A = $\left\{ 1,2,3,4,5 \right\}$ , B = $\left\{ 4,5,6,7,8 \right\}$ , C = $\left\{ 7,8,9,10,11 \right\}$ and D = $\left\{ 10,11,12,13,14 \right\}$ find:
(a). $A\cup B$
(b). $B\cup C$
(c). $A\cup C$
(d). $B\cup D$
(e). $\left( A\cup B \right)\cup C$
(f). $\left( A\cup B \right)\cap C$
(g). $\left( A\cap B \right)\cup D$
(h). $\left( A\cap B \right)\cup \left( B\cap C \right)$
(i). $\left( A\cup C \right)\cap \left( C\cup D \right)$

Answer 1

Hint: We will first give the definition of union and intersection and then we will substitute the value of A, B, C, D in the respective options and then perform the given operation to find the value of the given options.

Complete step-by-step answer:

Union: The union (denoted by $\cup $ ) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other.
Intersection: The intersection of two sets has only the elements common to both sets. If an element is in just one set it is not part of the intersection. The symbol is an upside down $\cap $ .
Now we will use this definition to solve the question.
For (a):
$A\cup B$= $\left\{ 1,2,3,4,5,6,7,8 \right\}$
Hence we have found the value of part (a).
For (b):
$B\cup C$ = $\left\{ 4,5,6,7,8,9,10,11 \right\}$
Hence we have found the value of part (b).
For (c):
$A\cup C$ = $\left\{ 1,2,3,4,5,7,8,9,10,11 \right\}$
Hence we have found the value of part (c).
For (d):
$B\cup D=\left\{ 4,5,6,7,8,10,11,12,13,14 \right\}$
Hence the value of part (d) has been found.
For (e):
$A\cup B$= $\left\{ 1,2,3,4,5,6,7,8 \right\}$
Now $\left( A\cup B \right)\cup C$ will be,
$\left( A\cup B \right)\cup C$ = $\left\{ 1,2,3,4,5,6,7,8,9,10,11 \right\}$
Hence we have found the value of part (e).
For (f):
$A\cup B$= $\left\{ 1,2,3,4,5,6,7,8 \right\}$
Now $\left( A\cup B \right)\cap C$ will be,
$\left( A\cup B \right)\cap C$ = $\left\{ 7,8 \right\}$
Hence we have found the value of part (f).
For (g):
$A\cap B=\left\{ 4,5 \right\}$
Now $\left( A\cap B \right)\cup D$ will be,
$\left( A\cap B \right)\cup D$ = $\left\{ 4,5,10,11,12,13,14 \right\}$
Hence we have found the value of part (g).
For (h):
$A\cap B=\left\{ 4,5 \right\}$
$B\cap C=\left\{ 7,8 \right\}$
Now $\left( A\cap B \right)\cup \left( B\cap C \right)$ will be,
$\left( A\cap B \right)\cup \left( B\cap C \right)$ = $\left\{ 4,5,7,8 \right\}$
Hence we have found the value of part (h).

For (i):
$A\cup C$ = $\left\{ 1,2,3,4,5,7,8,9,10,11 \right\}$
$C\cup D=\left\{ 7,8,9,10,11,12,13,14 \right\}$
Now $\left( A\cup C \right)\cap \left( C\cup D \right)$ will be,
$\left( A\cup C \right)\cap \left( C\cup D \right)$ = $\left\{ 7,8,9,10,11 \right\}$
Hence we have found the value of part (i).

Note: We have used the definition of intersection and union to solve this question. We can also solve this question by drawing a Venn diagram and then find all the required values. We must be careful and solve each part in an orderly manner and not get confused between union and intersection of sets. If by mistake, we find union of sets instead of intersection of sets and vice versa, then we might get completely different results.