Math Is Fun Forum

ISAwilla

Guest

how to calculate these intersections without writing all combinations

We have the following sets:

X=(a,b,c,d)∈S:b<c<d,
Y=(a,b,c,d)∈S:a<c<d,
Z=(a,b,c,d)∈S:a<b<d,
F=(a,b,c,d)∈S:a<b<c,
Where each of a,b,c,d have integer values from 1 to 5

How to calculate |X∩Y|, |X∩Z|, |Z∩F|, |X∩Y∩Z|, |X∩Y∩Z∩F| without the need to write down all possible combinations

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: how to calculate these intersections without writing all combinations

hi ISAwilla

Welcome to the forum.

By |X∩Y|, I assume you mean the number of elements in X∩Y

As a, b, c and d are numbers, it's hard to see how you could answer this without, at least, some consideration of possible combinations.

For X∩Y and subject to the numerical constraints, d must be the largest, then c, then either a or b.

So if d = 3 and a=b, there's just one case.

If d = 4 then we can find five cases:
when a < b
when b < a
when a = b (three here)

If d = 5
           then when c = 2, there is one case
           then when c = 3
                                  then when a < b, we have one case
                                          when b < a, we have one case
                                          when a = b, we have two cases
            then when c = 4
                                   then when a < b, we have three cases
                                   then when b < a, we have three cases
                                   then when a = b, we have three cases.

I cannot think how to get this without doing something like this.

Bob

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