How to find cases of overlap/not in overlap , mathematically

[rajemessage]

hi,

I have four variables (x and y),(a and b). i want to find out number of state which these variables can attain.

Example :
1) x=2 ,y =5 , a= 0 , b=2 ( that is a and b is less than x and y)
2) x=2 ,y =5 , a= 0 , b=2 ( that is a is less than x and b equal to x)
3) x=2 ,y =5 , a= 0 , b=3 ( that is a is less than x and b greater than x and less than y)
4) x=2 ,y =5 , a= 3 , b=4 ( that is a and be is in side x and y)
etc.
etc.
q1) is there any mathematical way to find all cases/state?

yours sincerley

 

[Ackbach]

There is. First of all, two variables can be in one of three states: x<y, x=y, x>y. This is called the Law of Trichotomy. Next, considering that we have four variables, how many relationships are there? Well, I count 6: x and y, x and a, x and b, y and a, y and b, a and b. There are six relationships, and each relationship can be one of three possibilities. So how many total possibilities are there?

 

[Evgeny.Makarov]

how many relationships are there? Well, I count 6: x and y, x and a, x and b, y and a, y and b, a and b. There are six relationships, and each relationship can be one of three possibilities.

I don't think all $3^6$ possibilities are realized. For example, if $x<y$ and $a<x$, then it can't be that $y<a$.

 

[Ackbach]

I don't think all $3^6$ possibilities are realized. For example, if $x<y$ and $a<x$, then it can't be that $y<a$.

Absolutely! I do think this is the place to start, though. You could list out those $729$ possibilities (quite a chore, really), and check each one for consistency. Hmm. Is there an easier way, do you think?

 

[Evgeny.Makarov]

I do think this is the place to start, though.

Frankly, I don't see how to use the idea that there are six pairs of variables, and each of them can be of at most three types. I may be missing something, though.

 

[rajemessage]

There is. First of all, two variables can be in one of three states: x<y, x=y, x>y. This is called the Law of Trichotomy. Next, considering that we have four variables, how many relationships are there? Well, I count 6: x and y, x and a, x and b, y and a, y and b, a and b. There are six relationships, and each relationship can be one of three possibilities. So how many total possibilities are there?

x<=y and a<=b