Intersecting Intervals: Consistency of Families F & G

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In summary, a consistent family is a set of intervals that intersect with each other. If two families, F and G, are both consistent and intersect with each other, then either F or G or both must also be consistent. This is true even when considering only the real number line. A counterexample for when neither F nor G is consistent is when F = {(1,5), (6,10)} and G = {(3,7), (2,8)}.
  • #1
Diffy
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I define a consistent family, F, to be a set of intervals such that if I_1 and I_2 are in F then I_1 and I_2 intersect

Now given two families F and G, we say F is consistent with G provided that each interval from F intersects with each interval from family G.

My question:
If you know that F is consistent with G, does this imply that either F or G is consistent itself?

My gut feeling is that if F is consistent with G then at either F OR G must be consistent..
 
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  • #2
Suppose F is a family of vertical intervals from (x, 1) to (x, -1) with x between -1 and 1 and G is a family of horizontal intervals from (1, y) to (-1, y) with y between -1 and 1.
 
  • #3
Hey Halls,
Great counterexample!

I am working in a book that is constructing the real number system through these intervals. As such, I have been strictly thinking of these rational intervals as one dimensional along the real number line.

If we restrict ourselves to the real number line then is my assertion true?
 
  • #4
Diffy said:
If we restrict ourselves to the real number line then is my assertion true?

Wee, I'd say, yes:smile:

Proof: edited...proof wrong
 
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  • #5
There must be something wrong Pere...
If I am reading your proof correctly you are asserting that both F and G are BOTH consistent. However consider F the family {(1,5), (6,10)} and G the family {(3,7), (2, 8)} Here F is consistent with G because each interval in F intersects each interval in G, however F is not consistent.
 
  • #6
You're right, my "proof" was dead wrong, gee ... I'll think of sth. else

Next try:

Assume F is consistent with G but neither F nor G is consistent. Clearly, F and G contain more than one interval, otherwise they would be consistent.
For two intervals v,w write v<w if [itex]\forall x\in v, \forall y\in w: x<w[/itex]. Our inconsistency assumption allows us to pick [itex]f_1, f_2[/itex] from F and [itex]g_1,g_2[/itex] from G such that

[tex]f_1\cap f_2 = \emptyset[/tex]
and
[tex]g_1\cap g_2 = \emptyset[/tex]

w.l.o.g. we take [itex]f_1<f_2[/itex] and [itex]g_1<g_2[/itex]. (If neither [itex]f_1<f_2[/itex] nor [itex]f_2<f_1[/itex] were true, there would be [itex]x, x' \in f_1,\quad y,y'\in f_2[/itex] such that x<y and x'>y'; this implies that the number
[tex]
t=\frac{y'(x-y)-y(x'-y')}{(x-y)-(x'-y')}
[/tex]
lies between x and x' as well as between y and y'.(I omit the calculation, it is rather obvious that one of two non-intersecting intervals has to be greater than the other one) Hence [itex]t\in f_1 \cap f_2[/itex] which is a contradiction.

F being consistent with G implies


[tex]f_1\cap g_2 :=h_2 \neq \emptyset[/tex]
[tex]f_2\cap g_1 :=h_3 \neq \emptyset[/tex]

Choose

[tex]x\in h_2,\quad y\in h_3[/tex]

There are now two cases:

Case 1: x<=y: Since [itex]x \in g_2[/itex] and [itex]y \in g_1[/itex] this is a contradiction to our assumption [itex]g_1<g_2[/itex].

Case 2: x>y: Since [itex]x \in f_1[/itex] and [itex]y \in f_2[/itex] this is a contradiction to our assumption [itex]f_1<f_2[/itex].

Hence, if F is consistent with G, either F or G or both have to be consistent.
 
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  • #7
Any objections to this proof?:smile:
 

Related to Intersecting Intervals: Consistency of Families F & G

1. What is the concept of intersecting intervals in relation to families F and G?

The concept of intersecting intervals in relation to families F and G refers to the consistency of these two families in terms of their intervals. This means that for any given interval in family F, there exists at least one interval in family G that shares a common point with it, and vice versa. In other words, the two families intersect at some point.

2. Why is the consistency of families F and G important in scientific research?

The consistency of families F and G is important in scientific research because it allows for more accurate and reliable results. When studying a particular phenomenon, it is crucial to have consistent and intersecting intervals in order to fully understand and analyze its characteristics and behavior.

3. How is the consistency of families F and G determined?

The consistency of families F and G is determined by analyzing the intervals of the two families and identifying any points where they intersect. This can be done through mathematical calculations or visual representations such as graphs or charts.

4. Can the consistency of families F and G be affected by external factors?

Yes, the consistency of families F and G can be affected by external factors such as changes in the environment or experimental conditions. These factors can alter the intervals of the two families and potentially disrupt their consistency.

5. What are the implications of inconsistent families F and G in scientific research?

Inconsistent families F and G can lead to inaccurate and unreliable results in scientific research. This can hinder the understanding and progress of a particular field of study, and potentially lead to incorrect conclusions and hypotheses.

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