Proving that the intersection of any two intervals is an interval

[artfullounger]

The question is as follows:

Prove that if I₁, I₂ are intervals and J = I₁$\cap$I₂ then J is an interval.

To be honest I don't even know where to start. There's a "hint" that suggests that I first write out the definitions of I₁, I₂, J as intervals and of the intersection between I₁ and I₂, but that hasn't really enlightened me...

So I just have:

A subset I_n of ℝ is an interval if $\forall$ x,y,z $\in$ ℝ , x$\in$I_n, z$\in$I_n and x<y<z then y$\in$I_n (I used n instead of 1 and 2 because I am too lazy to write it out twice. Also, substitute J in as appropriate xD )

I₁$\cap$I₂ = {x: x$\in$I₁ and x$\in$I₂}

I don't know where to go from there basically. If someone could even so much as nudge me in the right direction I would be very appreciative :D

 

[Deveno]

how would you go about deciding if x was in (a,b)∩(c,d)?

i'll give an example, which you can play with:

what is (2,4)∩(3,5)? that is:

if 2 < x < 4 AND

3 < x < 5, can you express just one inequality that covers both at once?

 

[artfullounger]

Hmm, but is it mathematically rigorous enough to, in a proof, say something along the lines of "given I₁ is the interval (I_1a,I_1b) and I₂ is the interval (I_2a,I_2b) , I₁$\cap$I₂
is the interval (I_1a,I_2b) and given J=I₁$\cap$I₂=and interval, J is an interval" ?

Because I don't have any definitions or theorems to work with that say anything remotely similar, at least that I'm aware of...

Actually you probably couldn't tell me whether that was sufficient seeing as you did not set the questions nor are you marking them... ¬.¬

 

[Deveno]

that's true...but, go ahead, answer the question of my earlier post:

what is (2,4)∩(3,5)?

 

[artfullounger]

2<x<5 ...right? Now I'm worried I've done something unbelievably stupid...

 

[Deveno]

try drawing a picture.

 

[artfullounger]

right, I HAVE done something unbelievably stupid then...so 3<x<4 ¬.¬ so I would actually ahve to revise my previous interval suggestion as (I_1b,I_2a)=J as well...

 

[Deveno]

hmm, it looks like 3 = max{2,3} and 4 = min{4,5}.

can you generalize?

(hint: suppose max{a,c} > min{b,d}. what is (a,b)∩(c,d) in this case).

 

[artfullounger]

well if the max of {a,c} > {b,d} then...would it be c>b, or would there be no interval?

as for generalizing...hmm...I have a kind of vague idea as to what I might say, btu I have no idea how to express it in a mathematical manner -_-

something along the lines of the interval being between greatest element of I₁ that is also $\in$I₂ and least element of I₂ that is also $\in$I₁

 

[Deveno]

i've given you a hint as to "the general formula".

suppose L = max{a,c} < M = min{b,d}.

try to describe (a,b)∩(c,d) in terms of L and M.

now, to consider "half-open" and "closed intervals", you'll have to do a careful case-by-case analysis. make sure you don't overlook the situation:

L = M.

 

[vela]

To show J is an interval, you want to show that if x, z ∈ J, then if x<y<z, then y ∈ J. The key thing here is to know what it means for a number to be in J.