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paulmdrdo1
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Understanding the Inclusion-Exclusion Property in Set Theory
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In summary, the conversation is discussing how to count elements in the union of three sets by using a formula that takes into account the intersections between the sets. The numbers inside the regions of circles represent the number of times that region is counted in the equation. The leftmost drawing shows the total number of elements in the union, the middle drawing shows the result of only using part of the formula, and the rightmost drawing shows the correct way to use the formula. The image is asking the viewer to solve the equation by determining the sets A, B, and C.
- #2
bergausstein
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paulmdrdo said:
does the numbers inside the regions of circles mean the cardinality of each set and intersections?
- #3
paulmdrdo1
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bergausstein said:
hi bergausstein! that's also my question.
:)- #4
I like Serena
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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.
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- #5
Mathelogician
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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
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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.
1. What is the Inclusion-Exclusion property?
The Inclusion-Exclusion property is a mathematical principle used to calculate the size of a set that is the union of two or more other sets. It states that the size of the union of two sets is equal to the sum of their sizes minus the size of their intersection.
2. How is the Inclusion-Exclusion property used in probability?
In probability, the Inclusion-Exclusion property is used to calculate the probability of events occurring together or separately. It allows us to calculate the probability of the union of two or more events by subtracting the probability of their intersection from the sum of their individual probabilities.
3. Can the Inclusion-Exclusion property be extended to more than two sets?
Yes, the Inclusion-Exclusion property can be extended to any number of sets. For example, the size of the union of three sets can be calculated by summing the sizes of each individual set, subtracting the sizes of their pairwise intersections, and then adding back the size of their triple intersection.
4. How is the Inclusion-Exclusion property used in combinatorics?
In combinatorics, the Inclusion-Exclusion property is used to count the number of elements in a set with certain properties. It allows us to take into account the elements that satisfy multiple properties without counting them twice. This is useful in problems involving counting arrangements, combinations, and permutations.
5. What is the significance of the Inclusion-Exclusion property?
The Inclusion-Exclusion property is a fundamental principle in mathematics and has applications in various fields such as probability, combinatorics, and set theory. It allows us to calculate the size or probability of complex sets by breaking them down into smaller, more manageable sets. It is also a useful tool for solving challenging counting and probability problems.
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