Prove that the set [1,3)={x∈R:1≤x<3} is an interval.
Also, prove that for any two intervals I , J , if I intersect J is not equal to ∅ then I ∪ J is an interval.
This is why I was getting confused because its a written up question not from a text and we were just given the formal definition of an interval from like back in calc.
A subset I of R is an interval provided that for all x, y, z ∈ R, if x < y < z and x ∈ I and z ∈ I then y ∈ I. (Given Def of Interval)
However, when I was looking through previous books they just jumped into open and closed intervals and nothing really about if the set is an interval. For the first part The set[1,3)={x∈R:1≤x<3}is an interval I would say its an interval but I don't know how to prove that it is. And then for the second part For any two intervals I , J , if I ∩ J ̸= ∅ then I ∪ J is an interval I'm not where to even start.
Let’s use that definition.
If $\displaystyle \{a,b\}\subset I\cup J~\&~a<c<b$ we must show that $\displaystyle c\in I\cup J$.
If we have either $\displaystyle \{a,b\}\subset I$ or $\displaystyle \{a,b\}\subset J$ then we are done because each of $\displaystyle I~\&~J$ is interval. WHY?
So suppose that $\displaystyle a\in I\setminus J$ and $\displaystyle b\in J\setminus I $. We are given that $\displaystyle \left( {\exists p \in I \cap J} \right)$.
Three cases: i) if $\displaystyle p=c$ we are done. Why?
ii) if $\displaystyle p<c<b$ because $\displaystyle J$ is an interval, so $\displaystyle c\in J\subset I\cup J$
iii) likewise if $\displaystyle a<c<p$ because $\displaystyle I$ is an interval, so $\displaystyle c\in I\subset I\cup J$.
We are done.