How Many Subset Pairs (X, Y) Meet These Mathematical Conditions?

[Lilia]

Mod note: Moved from a technical math forum, so missing the homework template.
Given a set A = {a₁, a₂, ..., a_n} and its two subsets - X, Y so that these subsets satisfy a condition, find the number of such possible (X, Y) pairs

Condition 1: X ∩ Y = {a₁, a₂},
Condition 2: X ⊆ Y,
Condition 3: | (X - Y) ∪ (Y - X) | = 1

I've tried to solve these but I still can't figure out the right way to think when solving these.

For X ∩ Y = ∅ the number of subsets is ΣC(n,k) × 2^n-k, where k=0÷n.

This is how I think - so, for Condition 1, X and Y should have only a₁ and a₂ in common and can have other distinct elements, so we need to choose those two elements but how to choose them for both X and Y?

For Condition 2, |X| can be k, |Y| can be k (or k-1?), therefore the number of subsets - C(n,k) × C(n,k) = (C(n,k))²? Is this right?

For Condition 3, X is a subset of Y, or Y is a subset of X, and they have one element in common, right? If so, how to count the number of subsets?

 

[Mark44]

Given a set A = {a₁, a₂, ..., a_n} and its two subsets - X, Y so that these subsets satisfy a condition, find the number of such possible (X, Y) pairs

Condition 1: X ∩ Y = {a₁, a₂},
Condition 2: X ⊆ Y,
Condition 3: | (X - Y) ∪ (Y - X) | = 1

I've tried to solve these but I still can't figure out the right way to think when solving these.

For X ∩ Y = ∅ the number of subsets is ΣC(n,k) × 2^n-k, where k=0÷n.

But Condition 1 says that X ∩ Y is not empty.

This is how I think - so, for Condition 1, X and Y should have only a₁ and a₂ in common and can have other distinct elements, so we need to choose those two elements but how to choose them for both X and Y?

a₁ and a₂ have to be elements of both X and Y.

For Condition 2, |X| can be k, |Y| can be k (or k-1?)

No, |Y| can't be k - 1.
If |Y| = k, then |X| can be at most k, but at least 2.
Condition 2 says that X is either a proper subset of Y or X = Y.

, therefore the number of subsets - C(n,k) × C(n,k) = (C(n,k))²? Is this right?

For Condition 3, X is a subset of Y, or Y is a subset of X, and they have one element in common, right? If so, how to count the number of subsets?

No, they have two elements in common, which is what Condition 1 says.
If X ⊂ Y, then X - Y = ∅, and Y - X would be all of the elements in Y that aren't elements of X
If X = Y, then X - Y = Y - X.

 

[Lilia]

Empty intersection wasn't a part of the problem, I just wrote that to show how I choose X and Y for a given condition (in that case - ∅)

For Condition 1, if I choose k elements for X (k≥2), then for Y I'll have to choose from n-k elements, that'll be ΣC(n,k) × 2^n-k, k=2÷n. How do I say that a₁ and a₂ and mutual for both X and Y?

For Condition 2, I just noticed I made a mistake, I should've said |X| is maybe k-1? So if it's a subset of Y then for Y we can choose C(n,k) and for X: 2^k

And for Condition 3, either X-Y or Y-X is ∅, that crossed my mind but I don't get why they have 2 elements in common? Condition 1 is a separate condition.

 

[fresh_42]

Condition 1 is a separate condition.

Does this mean, it consists of three separate exercises instead of all conditions have to hold simultaneously?
That makes a big difference which you haven't mentioned!

 

[Lilia]

All of the three are separate conditions, I'm sorry not to have mentioned it before

 

[Mark44]

@Lilia, in the future, please post homework-type questions in the Homework & Coursework sections, not in the technical math sections. I have moved your post.

 

[Mark44]

For Condition 1, I would investigate the problem with specific numbers for the number of elements in A, such as n = 2, n = 3, n = 4, and look for a pattern.

 

[Lilia]

Okay, got it.

For Condition 1, I got ΣC(n,k+2) × 2^n-k. Do I also need multiply this by C(n,2) and will that be correct?

 

[Ray Vickson]

Mod note: Moved from a technical math forum, so missing the homework template.
Given a set A = {a₁, a₂, ..., a_n} and its two subsets - X, Y so that these subsets satisfy a condition, find the number of such possible (X, Y) pairs

Condition 1: X ∩ Y = {a₁, a₂},
Condition 2: X ⊆ Y,
Condition 3: | (X - Y) ∪ (Y - X) | = 1

I've tried to solve these but I still can't figure out the right way to think when solving these.

For X ∩ Y = ∅ the number of subsets is ΣC(n,k) × 2^n-k, where k=0÷n.

This is how I think - so, for Condition 1, X and Y should have only a₁ and a₂ in common and can have other distinct elements, so we need to choose those two elements but how to choose them for both X and Y?

For Condition 2, |X| can be k, |Y| can be k (or k-1?), therefore the number of subsets - C(n,k) × C(n,k) = (C(n,k))²? Is this right?

For Condition 3, X is a subset of Y, or Y is a subset of X, and they have one element in common, right? If so, how to count the number of subsets?

If $X \subset Y$, do we not have $X \cap Y = X$?

 

[Lilia]

If $X \subset Y$, do we not have $X \cap Y = X$?

The three conditions are separate conditions, it's X ⊆ Y, so |X| can be anything from 0 to n (for n elements), and |Y| can be at most k, so I wrote ΣC(n,k) × 2ⁿ but does this ensure that the chosen Y subset contains X?

And for Condition 1 I should've written ΣC(n,k+2) × 2^n-k-2, where k=0,n-2. Or C(n,2) × (∑C(n-2,k) × 2^n-2-k), k=0,n-2?

 

[haruspex]

for Condition 1, X and Y should have only a1 and a2 in common and can have other distinct elements, so we need to choose those two elements but how to choose them for both X and Y?

Setting aside the two elements we know both must have, what conditions apply to the rest of X and Y?

For Condition 2, |X| can be k, |Y| can be k (or k-1?),

No, X is a subset of Y, Y cannot be smaller. If |X|=k, how many ways are there of choosing Y?

For Condition 3, X is a subset of Y, or Y is a subset of X, and they have one element in common, right? If so, how to count the number of subsets?

The first statement is right, but it is not that they have one element in common. Can you draw a Venn diagram?

 

[Lilia]

Setting aside the two elements we know both must have, what conditions apply to the rest of X and Y?

No, X is a subset of Y, Y cannot be smaller. If |X|=k, how many ways are there of choosing Y?

The first statement is right, but it is not that they have one element in common. Can you draw a Venn diagram?

Assuming that X-Y is empty, then drawing the Venn diagram - Y is a set (a circle) and X is another set (a circle) fully inside it.

And, then how many elements do they have in common?

 

[haruspex]

how many elements do they have in common

That's not the first question to ask here. How many elements do they have NOT in common?

 

[Lilia]

That's not the first question to ask here. How many elements do they have NOT in common?

That will be A-Y elements

 

[haruspex]

That will be A-Y elements

This is for condition 3, right? Maybe I need to be clearer. How many elements are in only one of X and Y?

 

[Lilia]

This is for condition 3, right? Maybe I need to be clearer. How many elements are in only one of X and Y?

Yes it's for the 3rd condition.
If there are k elements in Y (k=0,n) and if X-Y is empty and Y-X isn't empty then Y has more elements than X so there are m=0,k-1 (or m=k-1?) elements in X?

 

[haruspex]

or m=k-1?

That is the interesting question. What do you think?
(If you are considering the case where X is the smaller, it might be simpler to work in terms of what is the size of Y given the size of X.)