Question

If A={a,b,c,d,e,f}, B={c,e,g,h} and C={a,e,m,n}. Then find
(i) $A\cup B$
(ii) $B\cup C$
(iii) $A\cup C$

Answer 1

Hint:In this question we are given the definition of three sets and we have to find the union of each possible pair of two sets from these three sets. So, first we should understand the definition of union of two sets and use that definition to find the answer to this question.

Complete step-by-step answer:
In set theory, a set is a well-defined collection of objects. Therefore, each set will contain some elements. The union of two sets is defined as the collection of all the elements which belong to either A or B where A and B are two sets. Mathematically, we can write
$A\cup B=\left\{ x:x\in A\text{ or }x\in B \right\}............(1.1)$
Where the symbol $\cup $ represents union and the symbol $\in $ stands for belongs to.
In this question, A is defined to be
A= { a,b,c,d,e,f }
B is defined to be
B= { c,e,g,h }
and C is defined to be
C={a,e,m,n}
(i) Using equation (1.1), we see that $A\cup B$ should contain all the elements which belong to either A or B. However, in representing a set, each element must be written only once. Thus,
$\begin{align}
  & A\cup B=\left\{ x:x\in A\text{ or }x\in B \right\} \\
 & \Rightarrow A\cup B=\left\{ a,b,c,d,e,f,g,h \right\}......(1.2) \\
\end{align}$
Which is the answer to part (i) of the question.
(ii) Using equation (1.1), we see that $B\cup C$ should contain all the elements which belong to either B or C. However, in representing a set, each element must be written only once. Thus,
$\begin{align}
  & B\cup C=\left\{ x:x\in B\text{ or }x\in C \right\} \\
 & \Rightarrow B\cup C=\left\{ a,c,e,g,h,m,n \right\}......(1.3) \\
\end{align}$
Which is the answer to part (ii) of the question.
(iii) Using equation (1.1), we see that $A\cup C$ should contain all the elements which belong to either A or C. However, in representing a set, each element must be written only once. Thus,
$\begin{align}
  & A\cup C=\left\{ x:x\in A\text{ or }x\in C \right\} \\
 & \Rightarrow A\cup C=\left\{ a,b,c,d,e,f,m,n \right\}......(1.4) \\
\end{align}$
Which is the answer to part (iii) of the question.

Note: In equation (1.2), (1.3) and (1.4), the elements of the sets have been written in alphabetical order. However, the order of the elements while representing a set does not matter and thus, the elements can be written in any order. However, care must be taken to write each element only once and avoid repetitions of the elements in the representation of the set.