- 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.
+ 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.
diff --git a/public/index.xml b/public/index.xml
index 3bc39a1..d7702d7 100644
--- a/public/index.xml
+++ b/public/index.xml
@@ -18,7 +18,7 @@
Wed, 20 Nov 2019 00:00:00 +0000https://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.
+ 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.
diff --git a/public/posts/eidma/index.html b/public/posts/eidma/index.html
index c337591..7425352 100644
--- a/public/posts/eidma/index.html
+++ b/public/posts/eidma/index.html
@@ -6,7 +6,7 @@
-
+
@@ -29,7 +29,7 @@
-
+
@@ -38,7 +38,7 @@
-
+
@@ -91,6 +91,10 @@
+
(\exists x \in \mathbb{X}) (\neg \phi(x)) \equiv \neg ( \{ p \in \mathbb{X} : \phi(p) \} = \mathbb{X} )
Order of quantifiers matters.
@@ -272,117 +280,117 @@
Cartesian product:
-
\(A \times B = \{ (p, q) : p \in A \land q \in B \}\)
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A \times B = \{ (p, q) : p \in A \land q \in B \}
-
Def: A relation \(R\) on a set \(\mathbb X\) is a subset of \(\mathbb X \times \mathbb X\) (\(R \subseteq \mathbb X \times \mathbb X\))
-
Graph of a function \(f()\): \(\{ (x, f(x) : x \in Dom(f) \}\)
+
Def: A relation R on a set \mathbb X is a subset of \mathbb X \times \mathbb X (R \subseteq \mathbb X \times \mathbb X)
+
Graph of a function f(): \{ (x, f(x) : x \in Dom(f) \}
Properties of:
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Reflexivity: \((\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x\)
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Symmetricity: \([ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]\)
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Transitivity: \((\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)\)
-
Antisymmetricity: \((\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)\)
+
Reflexivity: (\forall x \in \mathbb X ) (x, x) \in R \equiv (\forall x \in \mathbb X) x R x
+
Symmetricity: [ (\forall x, y \in \mathbb X) (x, y) \in R \implies (y, x) \in R) ] \equiv [ (\forall x, y \in \mathbb X) ( x R y \implies y R x) ]
+
Transitivity: (\forall x, y, z \in \mathbb X) (x R y \land y R z \implies x R z)
+
Antisymmetricity: (\forall x, y \in \mathbb X) (x R y \land y R x \implies x = y)
Equivalence relations:
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Def: \(R \subseteq \mathbb X \times \mathbb X\) is said to be an equivalence relation iff\(R\) is reflexive, symmetric and transitive.
-
Congruence modulo n: \(p R q \equiv n | p - q\)
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Def R - and equivalence relation of \(\mathbb X\): The equivalence class of an element \(x \in \mathbb X\) is the set \([x]_R = \{ y \in \mathbb X : x R y \}\)
+
Def: R \subseteq \mathbb X \times \mathbb X is said to be an equivalence relation iffR is reflexive, symmetric and transitive.
+
Congruence modulo n: p R q \equiv n | p - q
+
Def R - and equivalence relation of \mathbb X: The equivalence class of an element x \in \mathbb X is the set [x]_R = \{ y \in \mathbb X : x R y \}
-
Every \(x \in \mathbb X\) belongs to the equivalence class of some element \(a\).
A partition is a set containing subsets of some set \(\mathbb X\) such that their collective symmetric difference equals \(\mathbb X\). A partition of is a set \(\{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \}\) such that:
+
A partition is a set containing subsets of some set \mathbb X such that their collective symmetric difference equals \mathbb X. A partition of is a set \{ A_i: i \in \mathbb I \land A_i \subseteq \mathbb X \} such that:
The quotient set: \(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)
+
The quotient set: \mathbb X / R = \{ [a] : a \in \mathbb X \}
Posets
Partial orders
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\(\mathbb X\) is a set, \(R \subseteq \mathbb X \times \mathbb X\)
-
Def: \(R\) is a partial order on \(\mathbb X\) iff \(R\) is:
+
\mathbb X is a set, R \subseteq \mathbb X \times \mathbb X
+
Def: R is a partial order on \mathbb X iff R is:
Reflexive
Antisymmetric
Transitive
-
Def: \(m \in \mathbb X\) is said to be:
+
Def: m \in \mathbb X is said to be:
-
maximal element in \((\mathbb X, \preccurlyeq)\) iff \((\forall a \in \mathbb X) m \preccurlyeq a \implies m = a\)
-
largest iff \((\forall a \in \mathbb X) (a \preccurlyeq m)\)
-
minimal iff \((\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)\)
-
smallest iff \((\forall a \in \mathbb X) (m \preccurlyeq a)\)
+
maximal element in (\mathbb X, \preccurlyeq) iff (\forall a \in \mathbb X) m \preccurlyeq a \implies m = a
+
largest iff (\forall a \in \mathbb X) (a \preccurlyeq m)
+
minimal iff (\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)
+
smallest iff (\forall a \in \mathbb X) (m \preccurlyeq a)
-
Def: A partial order \(R\) on \(\mathbb X\) is said to be “total” iff \((\forall x, y \in \mathbb X) (x R y \lor y R x)\)
-
Def: A subset \(B\) of \(\mathbb X\) is called a chain “chain” iff \(B\) is totally ordered by \(R\)
+
Def: A partial order R on \mathbb X is said to be “total” iff (\forall x, y \in \mathbb X) (x R y \lor y R x)
+
Def: A subset B of \mathbb X is called a chain “chain” iff B is totally ordered by R
-
\(C(\mathbb X)\) - the set of all chains in \((\mathbb X, R)\)
-
A chain \(D\) in \((\mathbb X, R)\) is called a maximal chain iff \(D\) is a maximal element in \((C(\mathbb X), R)\)
-
\(K \subseteq \mathbb X\) is called an antichain in \((\mathbb X, R)\) iff \((\forall p, q \in K) (p R q \lor q R p \implies p = q)\)
-
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
+
C(\mathbb X) - the set of all chains in (\mathbb X, R)
+
A chain D in (\mathbb X, R) is called a maximal chain iff D is a maximal element in (C(\mathbb X), R)
+
K \subseteq \mathbb X is called an antichain in (\mathbb X, R) iff (\forall p, q \in K) (p R q \lor q R p \implies p = q)
+
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:
+
If \phi is a propositional function defined on \mathbb N, if:
-
\(\phi(1)\)
-
\((\forall n \geq 1) \phi(n) \implies \phi(n+1)\)
-
\((\forall k \geq 1) \phi(k)\)
+
\phi(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:
+
f: \mathbb X \to \mathbb Y
+
Def: f \subseteq \mathbb X \times \mathbb Y is said to be a function if:
-
\((\forall x \in \mathbb X)(\exists y \in \mathbb Y) (x, y) \in f(y = f(x))\)
-
\((\forall a \in \mathbb X)(\forall p, q \in \mathbb Y)((a, p) \in f \land (a, q) \in f \implies p = q)\)
+
(\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 \(f: \mathbb X \to \mathbb Y\):
+
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 \(f\)
+
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)
+
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: \(f^{-1}(\mathbb A)\)
+
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 \(\mathbb A\)
+
|\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: \[
+
If \mathbb A, \mathbb B \subseteq \mathbb X and |\mathbb A|, |\mathbb B| \in \mathbb N and \mathbb A \cap \mathbb B = \emptyset then |\mathbb A \cup \mathbb B| = |\mathbb A| + |\mathbb B|
+
Can be generalized as:
(\forall n ) \mathbb{A}_1, \mathbb{A}_2, ..., \mathbb{A}_n \in \mathbb{X} \land \\
|\mathbb{A}_1|, |\mathbb{A}_2|, ..., |\mathbb{A}_n| \in \mathbb{N} \implies \\
(\forall i, j \in \{1, 2, ..., n \})(i \neq j \implies \mathbb{A}_i \cap \mathbb{A}_j = \emptyset)
-\]
- 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.
+ 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.
diff --git a/public/posts/index.xml b/public/posts/index.xml
index 737366e..da049dc 100644
--- a/public/posts/index.xml
+++ b/public/posts/index.xml
@@ -18,7 +18,7 @@
Wed, 20 Nov 2019 00:00:00 +0000https://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.
+ 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.
diff --git a/public/tags/index.html b/public/tags/index.html
index 709f098..ddcb4c6 100644
--- a/public/tags/index.html
+++ b/public/tags/index.html
@@ -93,6 +93,10 @@
+
- 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.
+ 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.
diff --git a/public/tags/university-notes/index.xml b/public/tags/university-notes/index.xml
index f8316aa..917dc34 100644
--- a/public/tags/university-notes/index.xml
+++ b/public/tags/university-notes/index.xml
@@ -18,7 +18,7 @@
Wed, 20 Nov 2019 00:00:00 +0000https://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.
+ 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.