From 4250b4575951f63c14dd0716baf1955786146bfb Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Abdulkadir=20Furkan=20=C5=9Eanl=C4=B1?= Date: Wed, 27 Nov 2019 15:10:32 +0100 Subject: [PATCH] Update eidma.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Abdulkadir Furkan Şanlı --- content/posts/eidma.md | 19 +++++++++++++++++++ 1 file changed, 19 insertions(+) diff --git a/content/posts/eidma.md b/content/posts/eidma.md index 408a2e9..b8fdaa4 100644 --- a/content/posts/eidma.md +++ b/content/posts/eidma.md @@ -179,3 +179,22 @@ markup = "pandoc" - Fact: $f^{-1}$ is a function iff $f$ is a *bijection* (1 to 1 and onto) - For some set $\mathbb A$ the image of $\mathbb A$ by $f$ is $f(\mathbb A) = \{ f(x) : x \in \mathbb A \}$. We can also define the inverse of an image even when the function itself isn't invertible: $f^{-1}(\mathbb A)$ + +## Combinatorics + +- $|\mathbb A|$ size (number of elements) of $\mathbb A$ +- Rule of addition: + - If $\mathbb A, \mathbb B \subseteq \mathbb X$ and $|\mathbb A|, |\mathbb B| \in \mathbb N$ and $\mathbb A \cap \mathbb B = \emptyset$ then $|\mathbb A \cup \mathbb B| = |\mathbb A| + |\mathbb B|$ + - Can be generalized as: +$$ +(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land \\ +|\mathbb{A}_1|, |\mathbb{A}_2|, ..., |\mathbb{A}_n| \in \mathbb{N} \implies \\ +(\forall i, j \in \{1, 2, ..., n \})(i \neq j \implies \mathbb{A}_i \cap \mathbb{A}_j = \emptyset) +$$ +- Rule of multiplication: + - $\mathbb{A}, \mathbb{B} \subseteq \mathbb{X}, |\mathbb{A} \times \mathbb{B}| = |\mathbb{A}| \cdot |\mathbb{B}|$ + - Can be generalized as: +$$ +(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land |\mathbb{A}_i| \in \mathbb{N} \implies \\ +|\mathbb{A}_1 \times \mathbb{A}_2 \times ... \times \mathbb{A}_n| = |\mathbb{A}_1| \cdot |\mathbb{A}_2| \cdot ... \cdot |\mathbb{A_n}| +$$