Thread: prove the set is an interval

1. prove the set is an interval

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.

2. Originally Posted by tn11631
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.
How does your text material define interval?

3. Originally Posted by Plato
How does your text material define 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.

4. Originally Posted by tn11631
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)
Let’s use that definition.
If $\{a,b\}\subset I\cup J~\&~a we must show that $c\in I\cup J$.
If we have either $\{a,b\}\subset I$ or $\{a,b\}\subset J$ then we are done because each of $I~\&~J$ is interval. WHY?

So suppose that $a\in I\setminus J$ and $b\in J\setminus I$. We are given that $\left( {\exists p \in I \cap J} \right)$.
Three cases: i) if $p=c$ we are done. Why?
ii) if $p because $J$ is an interval, so $c\in J\subset I\cup J$
iii) likewise if $a because $I$ is an interval, so $c\in I\subset I\cup J$.

We are done.