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title TARGET DECK FILE TAGS tags
Principle of Inclusion/Exclusion Obsidian::STEM combinatorics set
combinatorics
set

Overview

The principle of inclusion/exclusion refers to the oscillating adding and subtracting used to find the cardinality of potentially overlapping sets. Consider sets A, B, and C. Then

|A \cup B| = |A| + |B| - |AB|

and

|A \cup B \cup C| = |A| + |B| + |C| - |AB| - |AC| - |BC| + |ABC|

Notice the number of terms containing one set, two sets, three sets, etc. match the combinations#Binomial Coefficients.

%%ANKI Basic Given finite sets A and B and using PIE, what is |A \cup B|? Back: |AB| = |A| + |B| - |AB| Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Given finite sets A and B, what combinatorial concept is used to find |A \cup B|? Back: The principle of inclusion/exclusion. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Why is the principle of inclusion/exclusion named the way it is? Back: Because it involves an alternating adding and subtracting of terms. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What concept does PIE refer to? Back: The principle of inclusion/exclusion. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Given finite sets A and B, what is |A \cup B \cup C|? Back: |A| + |B| + |C| - |AB| - |AC| - |BC| + |ABC| Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Using sigma notation, what binomial identity is used to prove PIE correctly counts members? Back: \sum_{k=0}^n (-1)^k \binom{n}{k} = 0 Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Why might PIE be considered a top-down approach to counting? Back: It starts by counting every member of each union operand. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic What is the bottom-up approach contrasting PIE? Back: Apply the additive property to all disjoint sets the union operands can make. Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

%%ANKI Basic Given finite sets A and B and using a bottom-up approach (i.e. not PIE), what is |A \cup B|? Back: |A \cup B| = |AB| + |A - AB| + |B - AB| Reference: Oscar Levin, Discrete Mathematics: An Open Introduction, 3rd ed., n.d., https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf.

END%%

References