Question

Let A= {x: x is a positive integer} and let B= {x: x is a negative integer}. Find $A\cup B$.

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= {x: x is a positive integer}
and B is defined to be
B= {x: x is a negative integer}
Therefore, using equation (1.1), we get
$\begin{align}
  & A\cup B=\left\{ x:x\in A\text{ or }x\in B \right\} \\
 & \Rightarrow A\cup B=\left\{ x:x\text{ is a positive integer or }x\text{ is a negative integer} \right\}......(1.2) \\
\end{align}$
Now, any integer has to be either positive or negative or zero. Therefore, from equation (1.2), we find that $A\cup B$ contains all integers except zero. Therefore, we can write
$A\cup B=Z\backslash \{0\}...........(1.3)$
Where Z represents the set of integers and \ is used to denote that an element has been removed from that set, i.e. $A\cup B$ contains all elements of Z except 0. In roster form, we can write
$A\cup B=\left\{ ...-2,-1,1,2... \right\}...........(1.4)$
Which is the answer to the given question.

Note: We should note that both (1.3) and (1.4) would be correct answers to the given questions because they are just different representations of the same set. Equation (1.3) is in the set-builder form whereas equation (1.4) is in the roster form.