From e6290c0bd22fd2c741f7403a0ce0522c80f280b0 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Abdulkadir=20Furkan=20=C5=9Eanl=C4=B1?= Date: Wed, 20 Nov 2019 11:07:13 +0100 Subject: [PATCH] Update eidma MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Signed-off-by: Abdulkadir Furkan Şanlı --- content/posts/eidma.mmark | 23 +++++++++++--- public/index.html | 5 ++- public/index.xml | 24 +++++++------- public/posts/eidma/index.html | 39 ++++++++++++++++++----- public/posts/index.html | 5 ++- public/posts/index.xml | 4 +-- public/sitemap.xml | 42 ++++++++++++------------- public/tags/index.xml | 4 +-- public/tags/university-notes/index.html | 5 ++- public/tags/university-notes/index.xml | 4 +-- 10 files changed, 94 insertions(+), 61 deletions(-) diff --git a/content/posts/eidma.mmark b/content/posts/eidma.mmark index d80aaae..bcef92b 100644 --- a/content/posts/eidma.mmark +++ b/content/posts/eidma.mmark @@ -1,10 +1,7 @@ +++ title = "Introduction to Discrete Mathematics" -date = "2019-11-04" -author = "Abdulkadir" -showFullContent = false +date = "2019-11-20" tags = ["university-notes"] -markup = "mmark" +++ - Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. @@ -161,7 +158,23 @@ markup = "mmark" - Def: $$R$$ is a partial order on $$\mathbb X$$, $$R$$ is called a *well* order iff $$R$$ is a total order on $$X$$ and every nonempty subset $$A$$ of $$\mathbb X$$ has the smallest element ## Induction + - If $$\phi$$ is a propositional function defined on $$\mathbb N$$, if: - $$\phi(1)$$ - - $$(\forall n \geq 1) (\phi(n) \implies \phi(n+1)$$ + - $$(\forall n \geq 1) \phi(n) \implies \phi(n+1)$$ - $$(\forall k \geq 1) \phi(k)$$ + +## Functions + +- $$f: \mathbb X \to \mathbb Y$$ +- Def: $$f \subseteq \mathbb X \times \mathbb Y$$ is said to be a function if: + - $$(\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))$$ + - $$(\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)$$ + +- Types of functions $$f: \mathbb X \to \mathbb Y$$: + - $$f$$ is said to be an injection ( 1 to 1 function) iff $$(\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)$$ + - $$f$$ is said to be a surjection (onto function) iff $$(\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y$$ + - If $$f^{-1}$$ is a function from $$\mathbb Y \to \mathbb X$$ then $$f^{-1}$$ is called the inverse function for $$f$$ + - 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: $$f^{-1}(\mathbb A)$$ diff --git a/public/index.html b/public/index.html index 16eba8c..a78eb79 100644 --- a/public/index.html +++ b/public/index.html @@ -141,10 +141,9 @@ Introduction to Discrete Mathematics
- 2019-11-04 + 2019-11-20 - :: - Abdulkadir +
diff --git a/public/index.xml b/public/index.xml index 8c873b8..e2efb78 100644 --- a/public/index.xml +++ b/public/index.xml @@ -7,11 +7,22 @@ Hugo -- gohugo.io en-us © Abdulkadir Furkan Şanlı 2019 - Mon, 04 Nov 2019 11:14:55 +0100 + Wed, 20 Nov 2019 00:00:00 +0000 + + Introduction to Discrete Mathematics + https://abdulocra.cy/posts/eidma/ + Wed, 20 Nov 2019 00:00:00 +0000 + + https://abdulocra.cy/posts/eidma/ + Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false. We should avoid variables when forming statements, as they may change the logical value. + \(2=7\) statement \(x=5\) not a statement In logic we do not use the equals sign, we use the equivalence sign \(\equiv\). + Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc. + + about https://abdulocra.cy/about/ @@ -21,16 +32,5 @@ name: Abdulkadir Furkan Şanlı handle: abdulocracy contact: email: my handle at disroot dot org irc (freenode): abdulocracy - - Introduction to Discrete Mathematics - https://abdulocra.cy/posts/eidma/ - Mon, 04 Nov 2019 00:00:00 +0000 - - https://abdulocra.cy/posts/eidma/ - Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false. We should avoid variables when forming statements, as they may change the logical value. - \(2=7\) statement \(x=5\) not a statement In logic we do not use the equals sign, we use the equivalence sign \(\equiv\). - Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc. - - \ No newline at end of file diff --git a/public/posts/eidma/index.html b/public/posts/eidma/index.html index 74894e5..8e0edc3 100644 --- a/public/posts/eidma/index.html +++ b/public/posts/eidma/index.html @@ -35,7 +35,7 @@ \(2=7\) statement \(x=5\) not a statement In logic we do not use the equals sign, we use the equivalence sign \(\equiv\). Logical values (booleans) are denoted by either 0 or 1 (or t, f, etc." /> - + @@ -51,7 +51,7 @@ - + @@ -134,14 +134,10 @@
- 2019-11-04 + 2019-11-20 - :: - Abdulkadir - -
@@ -431,9 +427,36 @@
  • \(\phi(1)\)
    • -
  • \((\forall n \geq 1) (\phi(n) \implies \phi(n+1)\)
    • +
  • \((\forall n \geq 1) \phi(n) \implies \phi(n+1)\)
  • \((\forall k \geq 1) \phi(k)\)
+ + +

Functions

+ +
    +
  • \(f: \mathbb X \to \mathbb Y\)
    • + +

  • Def: \(f \subseteq \mathbb X \times \mathbb Y\) is said to be a function if:

    + +
      +
    • \((\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))\)
      • +
    • \((\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)\)
      • +
    • + +

  • Types of functions \(f: \mathbb X \to \mathbb Y\):

    + +
      +
    • \(f\) is said to be an injection ( 1 to 1 function) iff \((\forall x_1, x_2 \in \mathbb X) x_1 \neq x_2 \implies f(x_1) \neq f(x_2)\)
      • +
    • \(f\) is said to be a surjection (onto function) iff \((\forall y \in \mathbb Y)(\exists x \in \mathbb X) f(x) = y\)
      • +
    • If \(f^{-1}\) is a function from \(\mathbb Y \to \mathbb X\) then \(f^{-1}\) is called the inverse function for \(f\) + +
        +
      • 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: \(f^{-1}(\mathbb A)\)

diff --git a/public/posts/index.html b/public/posts/index.html index 5a604c1..1e0b5b0 100644 --- a/public/posts/index.html +++ b/public/posts/index.html @@ -138,10 +138,9 @@ Introduction to Discrete Mathematics
- 2019-11-04 + 2019-11-20 - :: - Abdulkadir +
diff --git a/public/posts/index.xml b/public/posts/index.xml index 3c03c69..35189eb 100644 --- a/public/posts/index.xml +++ b/public/posts/index.xml @@ -7,7 +7,7 @@ Hugo -- gohugo.io en-us © Abdulkadir Furkan Şanlı 2019 - Mon, 04 Nov 2019 00:00:00 +0000 + Wed, 20 Nov 2019 00:00:00 +0000 @@ -15,7 +15,7 @@ Introduction to Discrete Mathematics https://abdulocra.cy/posts/eidma/ - Mon, 04 Nov 2019 00:00:00 +0000 + Wed, 20 Nov 2019 00:00:00 +0000 https://abdulocra.cy/posts/eidma/ Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false. We should avoid variables when forming statements, as they may change the logical value. diff --git a/public/sitemap.xml b/public/sitemap.xml index be6ba31..9728fee 100644 --- a/public/sitemap.xml +++ b/public/sitemap.xml @@ -4,7 +4,27 @@ https://abdulocra.cy/ - 2019-11-04T11:14:55+01:00 + 2019-11-20T00:00:00+00:00 + + + + https://abdulocra.cy/posts/eidma/ + 2019-11-20T00:00:00+00:00 + + + + https://abdulocra.cy/posts/ + 2019-11-20T00:00:00+00:00 + + + + https://abdulocra.cy/tags/ + 2019-11-20T00:00:00+00:00 + + + + https://abdulocra.cy/tags/university-notes/ + 2019-11-20T00:00:00+00:00 @@ -12,26 +32,6 @@ 2019-11-04T11:14:55+01:00 - - https://abdulocra.cy/posts/eidma/ - 2019-11-04T00:00:00+00:00 - - - - https://abdulocra.cy/posts/ - 2019-11-04T00:00:00+00:00 - - - - https://abdulocra.cy/tags/ - 2019-11-04T00:00:00+00:00 - - - - https://abdulocra.cy/tags/university-notes/ - 2019-11-04T00:00:00+00:00 - - https://abdulocra.cy/categories/ diff --git a/public/tags/index.xml b/public/tags/index.xml index f6f56b3..6b9e327 100644 --- a/public/tags/index.xml +++ b/public/tags/index.xml @@ -7,7 +7,7 @@ Hugo -- gohugo.io en-us © Abdulkadir Furkan Şanlı 2019 - Mon, 04 Nov 2019 00:00:00 +0000 + Wed, 20 Nov 2019 00:00:00 +0000 @@ -15,7 +15,7 @@ university-notes https://abdulocra.cy/tags/university-notes/ - Mon, 04 Nov 2019 00:00:00 +0000 + Wed, 20 Nov 2019 00:00:00 +0000 https://abdulocra.cy/tags/university-notes/ diff --git a/public/tags/university-notes/index.html b/public/tags/university-notes/index.html index 3936bd9..86e02ee 100644 --- a/public/tags/university-notes/index.html +++ b/public/tags/university-notes/index.html @@ -138,10 +138,9 @@ Introduction to Discrete Mathematics
- 2019-11-04 + 2019-11-20 - :: - Abdulkadir +
diff --git a/public/tags/university-notes/index.xml b/public/tags/university-notes/index.xml index 8dce9c3..2a980c2 100644 --- a/public/tags/university-notes/index.xml +++ b/public/tags/university-notes/index.xml @@ -7,7 +7,7 @@ Hugo -- gohugo.io en-us © Abdulkadir Furkan Şanlı 2019 - Mon, 04 Nov 2019 00:00:00 +0000 + Wed, 20 Nov 2019 00:00:00 +0000 @@ -15,7 +15,7 @@ Introduction to Discrete Mathematics https://abdulocra.cy/posts/eidma/ - Mon, 04 Nov 2019 00:00:00 +0000 + Wed, 20 Nov 2019 00:00:00 +0000 https://abdulocra.cy/posts/eidma/ Mathematics without infinitely small, continuous mathematical objects. The mathematics of finite sets. Propositional calculus Comes from the linguistic concept that things can be either true or false. We should avoid variables when forming statements, as they may change the logical value.