I am trying to complete practice questions for my exam and have come across some set problems that I honestly have no idea how to answer. I am of course familiar with symbols for the universal set, natural numbers etc, I just don't recall my lecturer ever teaching us how to write such sets in listing notation! I have never used such notation. I would greatly appreciate any help I could get for the following questions which ask to write each set in listing notation:
- $U= \{n \in \mathbb{Z} : |n-7| < \pi\}$
- $A= \{n \in U : \text{$n$ is even}\}$
- $C= A \cup B$
Here's my ideas so far as to how I may go about the question:
I know that U stands for the universal set and its trying to say "the natural numbers in $\mathbb{Z}$ (integer numbers) such that .........now I really don't know how to incorporate this set of absolute values into the set. It's confusing me even further that $\pi$ is within this question because it isnt a natural or integer number!
Maybe I would write $A= \{2, 4, 6, 8, \dots\}$?
I'm familiar with set notation and understand that the symbol in the middle of A and B represents the "union" of the two but I don't how to go about such a question without context. All I can think of would be to do the following: C= (A,B)
Find all integers $n$ such that $|n-7|\lt\pi.$ First let's rewrite the inequality: $$|n-7|\lt\pi\iff-\pi\lt n-7\lt\pi\iff7-\pi\lt n\lt7+\pi.$$ In other words, $n$ lies in the interval $(7-\pi,7+\pi),$ or approximately $(3.86,10.14).$ The integers in that interval are $4,5,6,7,8,9,10.$ Thus $$U=\{4,5,6,7,8,9,10\}.$$
$A=\{n\in U:n\text{ is even}\}=\{4,6,8,10\}.$
$C=A\cup B=\{4,6,8,10\}\cup B=$?? What is $B$?