Add second year
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- #[[MA284 - Discrete Mathematics]]
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- **Previous Topic:** [[Counting]]
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- **Next Topic:** <tba>
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- **Relevant Slides:** 
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-
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- (Aside) **Notation for Complement:** $A^C$ for some set $A$.
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- ## Counting with Sets
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- ### Additive Principle for Sets
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- #### Additive Principle in terms of "events"
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- 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.
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- But, an "event" can just be expressed as ^^selecting an element of a set.^^
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- Saying "Event $A$ can occur $m$ ways" is ^^the same^^ as saying "$|A|=m$".
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- Saying "Event $B$ can occur $n$ ways is the same as saying "$|B|=n$".
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- If events $A$ and $B$ are **disjoint** / independent, that means that $|A \cap B| = 0$, or, equivalently , $A \cap B = \emptyset$.
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-
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- What is the **Additive Principle for Sets**? #card
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card-last-interval:: 4
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card-repeats:: 1
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card-ease-factor:: 2.36
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card-next-schedule:: 2022-09-18T19:24:25.898Z
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card-last-reviewed:: 2022-09-14T19:24:25.898Z
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card-last-score:: 3
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- Given two sets $A$ and $B$ with $|A \cap B| = 0$, then:
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- $$|A \cup B| = |A| + |B|$$
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- ### Multiplicative Principle for Sets
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- What is the **Cartesian Product** of two sets? #card
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card-last-interval:: 10.24
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card-repeats:: 3
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card-ease-factor:: 2.56
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card-next-schedule:: 2022-09-29T22:50:05.793Z
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card-last-reviewed:: 2022-09-19T17:50:05.793Z
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card-last-score:: 5
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- The **Cartesian Product** of two sets, $A$ and $B$ is:
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- $$A \times B = \{(x,y) : x \in A \text{ and } y \in B \}$$
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- 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 $B$.
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- #### Multiplicative Principle in terms of "events"
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- 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.
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- What is the **Multiplicative Principle for Sets**? #card
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card-last-interval:: 4
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card-repeats:: 2
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card-ease-factor:: 2.7
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card-next-schedule:: 2022-09-23T18:34:12.562Z
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card-last-reviewed:: 2022-09-19T18:34:12.563Z
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card-last-score:: 5
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- Given two sets, $A$ and $B$:
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- $$|A \times B| = |A| \cdot |B|$$
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- This extends to 3 or more sets in the obvious way.
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- ## The Principle of Inclusion & Exclusion (PIE)
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- What is the **Principle of Inclusion & Exclusion**? #card
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card-last-interval:: 4
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card-repeats:: 2
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card-ease-factor:: 2.7
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card-next-schedule:: 2022-09-30T12:18:17.188Z
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card-last-reviewed:: 2022-09-26T12:18:17.188Z
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card-last-score:: 5
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- For any two finite sets $A$ and $B$:
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- $$|A \cup B| = |A| + |B| - |A \cap B|$$
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- This also extends to larger numbers of sets, for example:
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- For any 3 finite sets $A$, $B$, and $C$:
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- $$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C |$$
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- ## Subsets & Power Sets
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- What is a **power set**? #card
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card-last-interval:: 4
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card-repeats:: 2
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card-ease-factor:: 2.46
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card-next-schedule:: 2022-09-23T17:49:45.527Z
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card-last-reviewed:: 2022-09-19T17:49:45.527Z
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card-last-score:: 5
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- The **power set** of a set $A$, denoted by $P(A)$, is the ^^set of all subsets^^ of $A$, ^^including the empty set^^ ($\emptyset$).
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-
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@ -0,0 +1,70 @@
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- #[[MA284 - Discrete Mathematics]]
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- **Previous Topic:** [[Counting]]
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- **Next Topic:** <tba>
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- **Relevant Slides:** 
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-
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||||
- (Aside) **Notation for Complement:** $A^C$ for some set $A$.
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- ## Counting with Sets
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- ### Additive Principle for Sets
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- #### Additive Principle in terms of "events"
|
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- 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.
|
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- But, an "event" can just be expressed as ^^selecting an element of a set.^^
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||||
- Saying "Event $A$ can occur $m$ ways" is ^^the same^^ as saying "$|A|=m$".
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- Saying "Event $B$ can occur $n$ ways is the same as saying "$|B|=n$".
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- If events $A$ and $B$ are **disjoint** / independent, that means that $|A \cap B| = 0$, or, equivalently , $A \cap B = \emptyset$.
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-
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- What is the **Additive Principle for Sets**? #card
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||||
card-last-interval:: 4
|
||||
card-repeats:: 1
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||||
card-ease-factor:: 2.36
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||||
card-next-schedule:: 2022-09-18T19:24:25.898Z
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||||
card-last-reviewed:: 2022-09-14T19:24:25.898Z
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card-last-score:: 3
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- Given two sets $A$ and $B$ with $|A \cap B| = 0$, then:
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- $$|A \cup B| = |A| + |B|$$
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- ### Multiplicative Principle for Sets
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- What is the **Cartesian Product** of two sets? #card
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||||
card-last-interval:: 10.24
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||||
card-repeats:: 3
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||||
card-ease-factor:: 2.56
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||||
card-next-schedule:: 2022-09-29T22:50:05.793Z
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||||
card-last-reviewed:: 2022-09-19T17:50:05.793Z
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card-last-score:: 5
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- The **Cartesian Product** of two sets, $A$ and $B$ is:
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- $$A \times B = \{(x,y) : x \in A \text{ and } y \in B \}$$
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- 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 $B$.
|
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- #### Multiplicative Principle in terms of "events"
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- 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.
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- What is the **Multiplicative Principle for Sets**? #card
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card-last-interval:: 4
|
||||
card-repeats:: 2
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||||
card-ease-factor:: 2.7
|
||||
card-next-schedule:: 2022-09-23T18:34:12.562Z
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||||
card-last-reviewed:: 2022-09-19T18:34:12.563Z
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||||
card-last-score:: 5
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||||
- Given two sets, $A$ and $B$:
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- $$|A \times B| = |A| \cdot |B|$$
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- This extends to 3 or more sets in the obvious way.
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- ## The Principle of Inclusion & Exclusion (PIE)
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||||
- What is the **Principle of Inclusion & Exclusion**? #card
|
||||
card-last-interval:: 4
|
||||
card-repeats:: 2
|
||||
card-ease-factor:: 2.7
|
||||
card-next-schedule:: 2022-09-30T12:18:17.188Z
|
||||
card-last-reviewed:: 2022-09-26T12:18:17.188Z
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||||
card-last-score:: 5
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- For any two finite sets $A$ and $B$:
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- $$|A \cup B| = |A| + |B| - |A \cap B|$$
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- This also extends to larger numbers of sets, for example:
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- For any 3 finite sets $A$, $B$, and $C$:
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- $$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C |$$
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- ## Subsets & Power Sets
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- What is a **power set**? #card
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card-last-interval:: 4
|
||||
card-repeats:: 2
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||||
card-ease-factor:: 2.46
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||||
card-next-schedule:: 2022-09-23T17:49:45.527Z
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||||
card-last-reviewed:: 2022-09-19T17:49:45.527Z
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||||
card-last-score:: 5
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||||
- The **power set** of a set $A$, denoted by $P(A)$, is the ^^set of all subsets^^ of $A$, ^^including the empty set^^ ($\emptyset$).
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-
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