# Thread:

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 $\{a,b\}\subset I\cup J~\&~a<c<b$ 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 $p<c<b$ because $J$ is an interval, so $c\in J\subset I\cup J$
iii) likewise if $a<c<p$ because $I$ is an interval, so $c\in I\subset I\cup J$.

We are done.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9352124333381653, "perplexity_flag": "head"}