- #1
paulmdrdo1
- 382
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Understanding the Inclusion-Exclusion Property in Set Theory
- Context: MHB
- Thread starter paulmdrdo1
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Discussion Overview
The discussion revolves around understanding the Inclusion-Exclusion property in set theory, specifically in the context of visual representations involving three sets A, B, and C. Participants are seeking clarification on how to interpret an image that illustrates the cardinalities of these sets and their intersections.
Discussion Character
- Exploratory
- Technical explanation
- Conceptual clarification
Main Points Raised
- Some participants express confusion about the meaning of the numbers inside the regions of the circles in the image, questioning if they represent the cardinality of each set and their intersections.
- One participant suggests that the image illustrates how to count elements in the union of the three sets, providing the formula for the Inclusion-Exclusion principle and explaining the counting process for overlapping elements.
- Another participant proposes a method for interpreting the image by assigning specific sets to the circles and calculating the total based on the cardinalities presented, emphasizing the importance of defining the sets first.
- A later reply reinforces the explanation of how the numbers indicate the count of elements in each region, detailing the process of double and triple counting and how to adjust for these counts using the Inclusion-Exclusion formula.
Areas of Agreement / Disagreement
Participants generally agree on the need to clarify the image and the role of the Inclusion-Exclusion principle, but there are varying interpretations of how to apply the concepts to the specific image in question. No consensus is reached on a single interpretation.
Contextual Notes
Some limitations in the discussion include missing context about the specific image being referenced and assumptions about the definitions of the sets involved. The mathematical steps and their implications remain unresolved.
- #2
bergausstein
- 191
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paulmdrdo said:
does the numbers inside the regions of circles mean the cardinality of each set and intersections?
- #3
paulmdrdo1
- 382
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bergausstein said:
hi bergausstein! that's also my question.
:)- #4
I like Serena
Science Advisor
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MHB
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Hi paulmdrdo!
Looks like some context is missing.
This looks like an explanation on how to count elements in the union of 3 sets A, B, and C.
Btw, the notation |A| means the number of elements in A (also called the cardinality of A).
The formula for that is:
$$|A\cup B \cup C| = |A|+|B|+|C|- \Big( |A \cap B|+|A \cap C| +|B \cap C| \Big)+|A \cap B \cap C|$$
In words: the total number of elements in the union is the sum of the elements in each set minus the elements in the mutual intersections plus the elements in the 3-way intersection.I believe the rightmost drawing represents the actual situation.
That is, we have 3 sets A, B, and C that each have 4 elements, such that each of the different types of intersections contain 1 element.
The total number of elements in the union is 7.
The leftmost drawing shows what you would count if you calculate |A|+|B|+|C|.
In that case each element in the intersections is counted twice, hence the 2 in the overlaps.
Except for the part where all 3 sets intersect where each element is counted thrice, hence the 3 in the middle.
The middle drawing represents what you get with only the part of the formula that is given below it.
The rightmost drawing is putting everything together, effectively showing the original situation.
Looks like some context is missing.
This looks like an explanation on how to count elements in the union of 3 sets A, B, and C.
Btw, the notation |A| means the number of elements in A (also called the cardinality of A).
The formula for that is:
$$|A\cup B \cup C| = |A|+|B|+|C|- \Big( |A \cap B|+|A \cap C| +|B \cap C| \Big)+|A \cap B \cap C|$$
In words: the total number of elements in the union is the sum of the elements in each set minus the elements in the mutual intersections plus the elements in the 3-way intersection.I believe the rightmost drawing represents the actual situation.
That is, we have 3 sets A, B, and C that each have 4 elements, such that each of the different types of intersections contain 1 element.
The total number of elements in the union is 7.
The leftmost drawing shows what you would count if you calculate |A|+|B|+|C|.
In that case each element in the intersections is counted twice, hence the 2 in the overlaps.
Except for the part where all 3 sets intersect where each element is counted thrice, hence the 3 in the middle.
The middle drawing represents what you get with only the part of the formula that is given below it.
The rightmost drawing is putting everything together, effectively showing the original situation.
Last edited:
- #5
Mathelogician
- 35
- 0
It seems it's the case that circles are filled up with numbers (look at numbers as objects like tomatoes!) and the image is asking you find solve the desired equations. Just you should decide at first the sets A, B and C in the image! unless you would not be able to solve it:). For example, for the leftmost picture, let the up-left circle be A, the up-right one be the set B, and the down one be C, the the answer of the equation |A|+|B|+|C| is equal to 4+4+4=12.
And the other possibility is what bergausstein mentioned! (And You again need to mark the sets first!)
And the other possibility is what bergausstein mentioned! (And You again need to mark the sets first!)
- #6
Jameson
- 4,533
- 13
This is just like I like Serena stated it.
The numbers indicate how many times that region has been counted using the equation below it. In the first one, you have three regions which are double counted and one region that is triple counted so you then subtract out $\Big( |A \cap B|+|A \cap C| +|B \cap C| \Big)$, which gets you close. We see that after doing this all regions are counted once, as desired, except for the middle region so we add back in $|A \cap B \cap C|$ and now we correctly are counting each region just one time.
The numbers indicate how many times that region has been counted using the equation below it. In the first one, you have three regions which are double counted and one region that is triple counted so you then subtract out $\Big( |A \cap B|+|A \cap C| +|B \cap C| \Big)$, which gets you close. We see that after doing this all regions are counted once, as desired, except for the middle region so we add back in $|A \cap B \cap C|$ and now we correctly are counting each region just one time.
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