Inclusion-Exclusion Quick Question

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Discussion Overview

The discussion revolves around the application of the Inclusion-Exclusion principle to determine the number of elements in exactly two out of four given sets. Participants explore different approaches and mathematical formulations related to this combinatorial problem.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Exploratory

Main Points Raised

  • One participant inquires about the approach to find the number of elements in exactly two of four sets, indicating a need for clarification on the problem's specifics.
  • Another participant provides a formula based on the Inclusion-Exclusion principle, suggesting that the interpretation of the sets is crucial for applying the formula correctly.
  • A third participant presents specific data for four sets, including cardinalities for individual sets and their intersections, and seeks guidance on how to calculate the number of elements in exactly two sets based on this data.
  • A fourth participant suggests using complements and intersections of the sets to find the desired cardinalities, although they express uncertainty about their own experience with the problem.

Areas of Agreement / Disagreement

Participants do not reach a consensus on a specific method or solution, and multiple approaches are proposed without resolution of which is preferable or correct.

Contextual Notes

Participants reference specific cardinalities and intersections, but the discussion lacks clarity on the definitions of the sets involved and the assumptions underlying the calculations.

Who May Find This Useful

Individuals interested in combinatorial mathematics, particularly those studying set theory and the Inclusion-Exclusion principle, may find this discussion relevant.

Caldus
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If I am given data for four sets and am asked to find how many elements are in exactly two of these four sets, how would I approach the problem?
 
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inc exc in this case states, if i let the sets be 1,2,3,4 that

|1u2u3u4| = |1|+|2|+|3|+|4| - |1n2|-|1n3|-|1n4|-|2n3|-|2n4|-|3n4|+|1n2n3| +|1n2n4|+|2n3n4|-|1n2n3n4|

how you use that depends on the details of the question. that is assuming that is what you meant by "4 sets", which 4 sets? which two of them do you need to find the cardinality of the interesection of?
 
If Universe = 75 and there for sets A1 - A4,

Each A has 28
Each intersection of two has 12 (Example: A1 n A2 = 12)
Each intersection of three has 5
Intersection of all sets equal 1

How do I go about finding how many elements are in exactly two sets?
 
Let Bi be the complement of Ai, then you want to find the cardinalities of all sets like

A1nA2mB3nB4

oughtn't to be too hard after you've played around with all the set identities you can think of, but to be honest, that I've not done.
 

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