Question

Let \[A=\text{ }\left\{ x:x=2n+1,n\in Z \right\}\]and let\[B=\text{ }\left\{ x:\text{ }x=2n,n\in Z \right\}\], the what is the value of $A\cup B$.

Answer 1

Hint:In this question we are given the definition of two sets and we have to find their union. 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. 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
\[A=\text{ }\left\{ x:x=2n+1,n\in Z \right\}\]
and B is defined to be
\[B=\text{ }\left\{ x:\text{ }x=2n,n\in Z \right\}\]
Therefore, using equation (1.1), we get
$\begin{align}
  & A\cup B=\left\{ x:x\in A\text{ or }x\in B \right\} \\
 & \Rightarrow A\cup B=\left\{ x:x=2n+1\text{ or }x=2n,n\in Z\text{ } \right\}......(1.2) \\
\end{align}$
Now, we may use the remainder theorem which states that for any two integers a and b, they can be written in the form
$a=nb+r..............(1.3)$
Where n and r are integers i.e. $n,r\in Z$ and ${r}<{b}$.
 Let x be any integer and let us take $a=x$ and $b=2$ in equation (1.3), therefore any integer x can be written can be written as
$x=n\times 2+r...........(1.4)$
However, as ${r}<{b}$, and here b=2, r can only take the values 0 and 1. Therefore, any integer can be written in the form
$x=2n+0\text{ or }x=2n+1............(1.5)$
Therefore, we find that any integer can be written in one of the forms stated in equation (1.5). Therefore, from equation (1.2) and equation (1.5), we find that $A\cup B$ contains all the integers. Therefore,
$Z\subseteq A\cup B...............(1.6)$
Also, as addition of integers and multiplication of integers gives only integers, we find that all the elements of $A\cup B$ should be integers
$A\cup B\subseteq Z..................(1.7)$
Now, from set theory if two sets M and N satisfy
$M\subseteq N\text{ and }N\subseteq M$ then $M=N$……………..(1.8)
Therefore, from equations (1.6) and (1.7), we find that
$A\cup B=Z.............(1.9)$
Where Z represents the set of integers. In roster form, we can write
$A\cup B=\left\{ ...-2,-1,0,1,2... \right\}............(2.0)$
Which is the answer to the given question.

Note: We should note that both (1.9) and (2.0) would be correct answers to the given questions because they are just different representations of the same set. Equation (1.9) is in the set-builder form whereas equation (2.0) is in the roster form.