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

  • Thread starter Lilia
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In summary, the problem requires finding the number of possible pairs of subsets (X, Y) from a set A with conditions that X and Y must satisfy. There are three separate conditions: X ∩ Y = {a1, a2} which means X and Y must have only a1 and a2 in common, X ⊆ Y which means X is a subset of Y, and | (X - Y) ∪ (Y - X) | = 1 which means X and Y have one common element. To solve this problem, one must find the number of possible subsets for each condition and then find the total number of possible pairs by combining them.
  • #1
Lilia
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Mod note: Moved from a technical math forum, so missing the homework template.
Given a set A = {a1, a2, ..., an} 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) × 2n-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?
 
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  • #2
Lilia said:
Given a set A = {a1, a2, ..., an} 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) × 2n-k, where k=0÷n.
But Condition 1 says that X ∩ Y is not empty.
Lilia said:
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.
Lilia said:
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.
Lilia said:
, 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.
 
  • #3
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) × 2n-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: 2k

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.
 
  • #4
Lilia said:
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!
 
  • #5
All of the three are separate conditions, I'm sorry not to have mentioned it before
 
  • #6
@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.
 
  • #7
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.
 
  • #8
Okay, got it.

For Condition 1, I got ΣC(n,k+2) × 2n-k. Do I also need multiply this by C(n,2) and will that be correct?
 
  • #9
Lilia said:
Mod note: Moved from a technical math forum, so missing the homework template.
Given a set A = {a1, a2, ..., an} 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) × 2n-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##?
 
  • #10
Ray Vickson said:
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) × 2n-k-2, where k=0,n-2. Or C(n,2) × (∑C(n-2,k) × 2n-2-k), k=0,n-2?
 
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  • #11
Lilia said:
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?
Lilia said:
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?
Lilia said:
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?
 
  • #12
haruspex said:
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?
 
  • #13
Lilia said:
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?
 
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  • #14
haruspex said:
That's not the first question to ask here. How many elements do they have NOT in common?
That will be A-Y elements
 
  • #15
Lilia said:
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?
 
  • #16
haruspex said:
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?
 
  • #17
Lilia said:
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.)
 

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

1. What is a subset pair?

A subset pair refers to a set of two subsets that are part of a larger set. For example, if the set is {1, 2, 3}, a subset pair could be {1, 2} and {1, 3}, as these are both subsets of the larger set.

2. How many subset pairs can be created from a set?

The number of subset pairs that can be created from a set is equal to 2 to the power of n, where n is the number of elements in the set. So, for a set with 3 elements, there would be 2 to the power of 3, or 8, subset pairs.

3. Can a subset pair contain duplicate elements?

No, a subset pair cannot contain duplicate elements. Each element in a subset pair must be unique and cannot be repeated.

4. How does the number of subset pairs change with different sizes of sets?

The number of subset pairs increases exponentially as the size of the set increases. For example, a set with 4 elements would have 16 subset pairs, while a set with 5 elements would have 32 subset pairs.

5. What is the significance of the number of subset pairs in mathematics?

The number of subset pairs has many applications in mathematics, particularly in combinatorics and probability. It is also used in various algorithms and data structures in computer science.

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