uni/year2/semester1/logseq-stuff/pages/Principle of Inclusion-Exclusion.md

• #MA284 - Discrete Mathematics
• Previous Topic: Counting
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• Relevant Slides:
• (Aside) Notation for Complement: A^C for some set A.

• Counting with Sets

  • Additive Principle for Sets

    • Additive Principle in terms of "events"

      • If Event A can occur m ways, and Event B can occur n (disjoint / independent) ways, then event "A or $B$" can occur in m+n ways.
      • But, an "event" can just be expressed as ^^selecting an element of a set.^^
        • Saying "Event A can occur m ways" is ^^the same^^ as saying "$|A|=m$".
        • Saying "Event B can occur n ways is the same as saying "$|B|=n$".
        • If events A and B are disjoint / independent, that means that |A \cap B| = 0, or, equivalently , A \cap B = \emptyset.
    • What is the Additive Principle for Sets? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.32 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T20:02:08.655Z card-last-score:: 1
      • Given two sets A and B with |A \cap B| = 0, then:
      • |A \cup B| = |A| + |B|

  • Multiplicative Principle for Sets

    • What is the Cartesian Product of two sets? #card card-last-interval:: 11.04 card-repeats:: 3 card-ease-factor:: 2.76 card-next-schedule:: 2022-11-25T16:36:56.463Z card-last-reviewed:: 2022-11-14T16:36:56.463Z card-last-score:: 5
      • The Cartesian Product of two sets, A and B is:
        • A \times B = \{(x,y) : x \in A \text{ and } y \in B \}
      • This is the set of pairs where the first term in each pair comes from A, and the second term in each pair comes from B.

    • Multiplicative Principle in terms of "events"

      • If event A can occur m ways, and each possibility allows for B to occur in n (disjoint) ways, then event "A and $B$" can occur in m \times n ways.
    • What is the Multiplicative Principle for Sets? #card card-last-interval:: 33.64 card-repeats:: 4 card-ease-factor:: 2.9 card-next-schedule:: 2022-12-18T11:00:28.884Z card-last-reviewed:: 2022-11-14T20:00:28.884Z card-last-score:: 5
      • Given two sets, A and B:
        • |A \times B| = |A| \cdot |B|
      • This extends to 3 or more sets in the obvious way.

• The Principle of Inclusion & Exclusion (PIE)

  • What is the Principle of Inclusion & Exclusion? #card card-last-interval:: 33.64 card-repeats:: 4 card-ease-factor:: 2.9 card-next-schedule:: 2022-12-18T11:01:03.942Z card-last-reviewed:: 2022-11-14T20:01:03.942Z card-last-score:: 5
    • For any two finite sets A and B:
      • |A \cup B| = |A| + |B| - |A \cap B|
    • This also extends to larger numbers of sets, for example:
      • For any 3 finite sets A, B, and C:
        • |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C |

• Subsets & Power Sets

  • What is a power set? #card card-last-interval:: 4 card-repeats:: 2 card-ease-factor:: 2.66 card-next-schedule:: 2022-11-27T12:19:42.366Z card-last-reviewed:: 2022-11-23T12:19:42.366Z card-last-score:: 5
    • The power set of a set A, denoted by P(A), is the ^^set of all subsets^^ of A, ^^including the empty set^^ (\emptyset).