Combinatorics with repetition

vault/Mathematics/Algebra/Algebraic Structures/Algebraic Structure.md

@@ -7,7 +7,7 @@ tags:
 
 >[!DEFINITION] Definition: Algebraic Structure
 >
->An **algebraic structure** $(S, O, R)$ consists of a [non-empty](../../Set%20Theory/The%20Empty%20Set.md) [set](../../Set%20Theory/index.md) $S$, a set $O$ of [operations](Operations/Operation.md) on $S$ and a finite set $R$ of rules which these operations must obey.
+>An **algebraic structure** $(S, O, R)$ consists of a [non-empty](../../Set%20Theory/The%20Empty%20Set.md) [set](../../Set%20Theory/Sets.md) $S$, a set $O$ of [operations](Operations/Operation.md) on $S$ and a finite set $R$ of rules which these operations must obey.
 >
 >>[!NOTE]
 >>

vault/Mathematics/Algebra/Algebraic Structures/Operations/Binary Operation.md

@@ -7,7 +7,7 @@ tags:
 
 >[!DEFINITION] Definition: Binary Operation
 >
->A **binary operation** on a [set](../../../Set%20Theory/index.md) $S$ is an [operation](Operation.md) $f: S \times S \to S$.
+>A **binary operation** on a [set](../../../Set%20Theory/Sets.md) $S$ is an [operation](Operation.md) $f: S \times S \to S$.
 >
 >>[!NOTATION]-
 >>

vault/Mathematics/Algebra/Algebraic Structures/Operations/Closure.md

@@ -7,9 +7,9 @@ tags:
 
 >[!DEFINITION] Definition: Closure
 >
->Let $S$ be a [set](../../../Set%20Theory/index.md) with an $n$-[ary operation](Operation.md) $o: \underset{n \text{ times}}{\underbrace{S \times \cdots \times S}} \to S$.
+>Let $S$ be a [set](../../../Set%20Theory/Sets.md) with an $n$-[ary operation](Operation.md) $o: \underset{n \text{ times}}{\underbrace{S \times \cdots \times S}} \to S$.
 >
->We say that a [subset](../../../Set%20Theory/index.md) $C \subseteq S$ is **closed** under $o$ if 
+>We say that a [subset](../../../Set%20Theory/Sets.md) $C \subseteq S$ is **closed** under $o$ if 
 >
 >$$
 >o(c_1, \cdots, c_n) \in C \qquad \forall c_1, \cdots, c_n \in C

vault/Mathematics/Algebra/Algebraic Structures/Operations/Operation.md

@@ -7,7 +7,7 @@ tags:
 
 >[!DEFINITION] Definition: Operation
 >
->An $n$**-ary operation** on a [set](../../../Set%20Theory/index.md) $S$ is a [function](../../../Analysis/Functions/index.md) from the $n$-ary [Cartesian power](../../../Set%20Theory/Set%20Operations.md) of $S$ to $S$.
+>An $n$**-ary operation** on a [set](../../../Set%20Theory/Sets.md) $S$ is a [function](../../../Analysis/Functions/index.md) from the $n$-ary [Cartesian power](../../../Set%20Theory/Set%20Operations.md) of $S$ to $S$.
 >
 >$$
 >f: \underset{n \text{ times}}{\underbrace{S \times \cdots \times S}} \to S

vault/Mathematics/Algebra/Algebraic Structures/Operations/Unary Operation.md

@@ -7,5 +7,5 @@ tags:
 
 >[!DEFINITION] Definition: Unary Operation
 >
->A **unary operation** on a [set](../../../Set%20Theory/index.md) $S$ is an [operation](Operation.md) $f: S \to S$.
+>A **unary operation** on a [set](../../../Set%20Theory/Sets.md) $S$ is an [operation](Operation.md) $f: S \to S$.
 >

vault/Mathematics/Algebra/Equations/Differential Equations/Ordinary Differential Equations/index.md

@@ -22,7 +22,7 @@ tags:
 >
 >>[!DEFINITION] Definition: Solution to an ODE
 >>
->>A **solution** to the ODE on the [subset](../../../../Set%20Theory/index.md) $\mathcal{D} \subseteq \mathbb{R}$ is an $n$-times [differentiable](../../../../Analysis/Real%20Analysis/Real%20Functions/Differentiation/Derivatives.md) [real function](../../../../Analysis/Real%20Analysis/Real%20Functions/Real%20Functions.md) $\phi: \mathcal{D} \to \mathbb{R}$ such that
+>>A **solution** to the ODE on the [subset](../../../../Set%20Theory/Sets.md) $\mathcal{D} \subseteq \mathbb{R}$ is an $n$-times [differentiable](../../../../Analysis/Real%20Analysis/Real%20Functions/Differentiation/Derivatives.md) [real function](../../../../Analysis/Real%20Analysis/Real%20Functions/Real%20Functions.md) $\phi: \mathcal{D} \to \mathbb{R}$ such that
 >>
 >>$$
 >>F\left(x, \phi(x), \phi'(x), \phi''(x), \dotsc, \phi^{(n)}(x)\right) = 0 \qquad \forall x \in \mathcal{D}

vault/Mathematics/Algebra/Equations/Equation.md

@@ -8,7 +8,7 @@ tags:
 
 >[!DEFINITION] Definition: Equation
 >
->Let $X$ and $Y$ be two not necessarily unique [sets](../../Set%20Theory/index.md) such that $Y$ has an [equivalence relation](../../Set%20Theory/Relations/Equivalence%20Relation.md) $E_Y$ defined on it.
+>Let $X$ and $Y$ be two not necessarily unique [sets](../../Set%20Theory/Sets.md) such that $Y$ has an [equivalence relation](../../Set%20Theory/Relations/Equivalence%20Relation.md) $E_Y$ defined on it.
 >
 >An **equation** over $X$ is an [expression](../../Formal%20Logic/Formal%20Languages.md) of the form
 >

vault/Mathematics/Algebra/Fields/The Complex Numbers/Operations.md

@@ -95,7 +95,7 @@ tags:
 
 >[!THEOREM] Theorem: The Field of the Complex Numbers
 >
->The [set](../../../Set%20Theory/index.md) of all [complex numbers](./index.md) $\mathbb{C}$ forms a [field](../index.md) together with the [addition and multiplication](Operations.md) defined on it.
+>The [set](../../../Set%20Theory/Sets.md) of all [complex numbers](./index.md) $\mathbb{C}$ forms a [field](../index.md) together with the [addition and multiplication](Operations.md) defined on it.
 >
 >>[!PROOF]-
 >>

vault/Mathematics/Algebra/Fields/The Complex Numbers/index.md

@@ -36,7 +36,7 @@ tags:
 >>
 >>Complex numbers are usually denoted by $z$.
 >>
->>The [set](../../../Set%20Theory/index.md) of all complex numbers is denoted by $\mathbb{C}$.
+>>The [set](../../../Set%20Theory/Sets.md) of all complex numbers is denoted by $\mathbb{C}$.
 >>
 >
 >>[!DEFINITION] Definition: Real Part

vault/Mathematics/Algebra/Fields/The Real Numbers/index.md

@@ -9,7 +9,7 @@ tags:
 ---
 >[!THEOREM] Theorem: The Field of the Real Numbers
 >
->The [set](../../../Set%20Theory/index.md) of the real numbers $\mathbb{R}$ together with the addition and multiplication operations defined on it form an [ordered field](../index.md).
+>The [set](../../../Set%20Theory/Sets.md) of the real numbers $\mathbb{R}$ together with the addition and multiplication operations defined on it form an [ordered field](../index.md).
 >
 >>[!PROOF]-
 >>

vault/Mathematics/Algebra/Linear Algebra/Linear Transformations/Dual Spaces/Dual Space.md

@@ -1,6 +1,6 @@
 >[!DEFINITION] Definition: Dual Space
 >
->The **dual space** of a [vector space](../../Vector%20Spaces/Vector%20Space.md) $(V,F,+,\cdot)$ is a [vector space](../../Vector%20Spaces/Vector%20Space.md) $(V^\ast, F, +, \cdot)$, where $V^\ast$ the [set](../../../../Set%20Theory/index.md) of all [linear forms](../../Linear%20Transformations/Linear%20Form.md) of $(V,F,+,\cdot)$ and the two operations $+: V^\ast \times V^\ast \to V^\ast$ and $\cdot: F\times V^\ast \to V^\ast$ are defined as
+>The **dual space** of a [vector space](../../Vector%20Spaces/Vector%20Space.md) $(V,F,+,\cdot)$ is a [vector space](../../Vector%20Spaces/Vector%20Space.md) $(V^\ast, F, +, \cdot)$, where $V^\ast$ the [set](../../../../Set%20Theory/Sets.md) of all [linear forms](../../Linear%20Transformations/Linear%20Form.md) of $(V,F,+,\cdot)$ and the two operations $+: V^\ast \times V^\ast \to V^\ast$ and $\cdot: F\times V^\ast \to V^\ast$ are defined as
 >
 >$$
 >\begin{align*} (\varphi + \psi)(x) &= \varphi(x) + \psi (x) \qquad \forall \varphi,\psi \in V^\ast, \forall x \in V \\ (a\varphi)(x) &= a\varphi(x) \qquad \forall \varphi \in V^\ast, \forall x \in V \end{align*}

vault/Mathematics/Algebra/Linear Algebra/Linear Transformations/Kernel.md

@@ -2,7 +2,7 @@
 >
 >Let $(V, F, +, \cdot)$ and $(W, F, +, \cdot)$ be [vector spaces](../Vector%20Spaces/Vector%20Space.md).
 >
->The **kernel** of a [linear transformation](Linear%20Transformation.md) $T: V \to W$ is the [set](../../../Set%20Theory/index.md) of all [vectors](../Vector%20Spaces/Vector.md) $\mathbf{v} \in V$ which the transformation sends to the zero vector in $W$:
+>The **kernel** of a [linear transformation](Linear%20Transformation.md) $T: V \to W$ is the [set](../../../Set%20Theory/Sets.md) of all [vectors](../Vector%20Spaces/Vector.md) $\mathbf{v} \in V$ which the transformation sends to the zero vector in $W$:
 >
 >$$
 >\ker (L) \overset{\text{def}}{=} \{\mathbf{v} \in V \mid T(\mathbf{v}) = \mathbf{0}_W\}

vault/Mathematics/Algebra/Linear Algebra/Matrices/Null Space.md

@@ -1,6 +1,6 @@
 >[!DEFINITION] Definition: Null Space
 >
->The **null space** of a [matrix](Matrix.md) $A \in F^{n \times n}$ is the [set](../../../Set%20Theory/index.md) of all [column vectors](Row%20and%20Column%20Vectors/Column%20Vector.md) $\vec{v} \in F^n$ which $A$ sends to the zero vector when [multiplying](Matrix%20Operations/Matrix%20Product.md) it by $\vec{v}$ on the right:
+>The **null space** of a [matrix](Matrix.md) $A \in F^{n \times n}$ is the [set](../../../Set%20Theory/Sets.md) of all [column vectors](Row%20and%20Column%20Vectors/Column%20Vector.md) $\vec{v} \in F^n$ which $A$ sends to the zero vector when [multiplying](Matrix%20Operations/Matrix%20Product.md) it by $\vec{v}$ on the right:
 >
 >$$
 >\{\vec{v}\in F^{n} \mid A\vec{v}=\vec{0}\}

vault/Mathematics/Algebra/Linear Algebra/Matrices/Square Matrices/Eigentheory/Eigenspace.md

@@ -2,7 +2,7 @@
 >
 >Let $A \in F^{n \times n}$ be a [square matrix](../Square%20Matrix.md).
 >
->The **eigenspace** of an [eigenvalue](Eigenvalue.md) $\lambda$ is the [set](../../../../../Set%20Theory/index.md) of all [eigenvectors](Eigenvector.md) which belong to $\lambda$ together with the [zero vector](../../Row%20and%20Column%20Vectors/Column%20Vector.md).
+>The **eigenspace** of an [eigenvalue](Eigenvalue.md) $\lambda$ is the [set](../../../../../Set%20Theory/Sets.md) of all [eigenvectors](Eigenvector.md) which belong to $\lambda$ together with the [zero vector](../../Row%20and%20Column%20Vectors/Column%20Vector.md).
 >
 >$$\{\vec{v} \in F^n \mid A\vec{v} = \lambda \vec{v}\}$$
 >

vault/Mathematics/Algebra/Linear Algebra/Matrices/Vector Space of Matrices.md

@@ -1,6 +1,6 @@
 >[!THEOREM] Theorem: Vector Space of Matrices
 >
->The [set](../../../Set%20Theory/index.md) $F^{m\times n}$ of all $m \times n$[-matrices](Matrix.md) and the [field](../../Fields/index.md) $F$ form a [vector space](../Vector%20Spaces/Vector%20Space.md) $(F^{m\times n}, F, +, \cdot)$ together with the [addition](Matrix%20Operations/Matrix%20Addition.md) and [scalar multiplication](Matrix%20Operations/Scalar%20Multiplication.md) for [matrices](Matrix.md).
+>The [set](../../../Set%20Theory/Sets.md) $F^{m\times n}$ of all $m \times n$[-matrices](Matrix.md) and the [field](../../Fields/index.md) $F$ form a [vector space](../Vector%20Spaces/Vector%20Space.md) $(F^{m\times n}, F, +, \cdot)$ together with the [addition](Matrix%20Operations/Matrix%20Addition.md) and [scalar multiplication](Matrix%20Operations/Scalar%20Multiplication.md) for [matrices](Matrix.md).
 >
 >>[!NOTE] Note: Zero Vector
 >>

vault/Mathematics/Algebra/Linear Algebra/Systems of Linear Equations/Solvability of a System of Linear Equations.md

@@ -4,7 +4,7 @@
 >
 >>[!DEFINITION] Definition: Solution Space
 >>
->>The **solution space** of a [system of linear equations](System%20of%20Linear%20Equations.md) is the [set](../../../Set%20Theory/index.md) of all [tuples](../../../Set%20Theory/Tuples.md) which are [solutions](Solvability%20of%20a%20System%20of%20Linear%20Equations.md) of the system.
+>>The **solution space** of a [system of linear equations](System%20of%20Linear%20Equations.md) is the [set](../../../Set%20Theory/Sets.md) of all [tuples](../../../Set%20Theory/Tuples.md) which are [solutions](Solvability%20of%20a%20System%20of%20Linear%20Equations.md) of the system.
 >>
 >
 >>[!DEFINITION] Definition: Solvability

vault/Mathematics/Algebra/Linear Algebra/Systems of Linear Equations/System of Linear Equations.md

@@ -1,6 +1,6 @@
 >[!DEFINITION] Definition: System of Linear Equations
 >
->A **system of** $m$ **linear equations** with $n$ unknowns $x_1, \cdots, x_n$ over some [field](../../Fields/index.md) $F$ is a [set](../../../Set%20Theory/index.md) of $m$ [equations](../../Equations/Equation.md) which can be expressed in the following way:
+>A **system of** $m$ **linear equations** with $n$ unknowns $x_1, \cdots, x_n$ over some [field](../../Fields/index.md) $F$ is a [set](../../../Set%20Theory/Sets.md) of $m$ [equations](../../Equations/Equation.md) which can be expressed in the following way:
 >
 >$$
 >\left|\begin{align*}a_{11}x_1 + \cdots + a_{1n}x_n &= b_1 \\ a_{21}x_1+\cdots+a_{2n}x_n &= b_2 \\\vdots \hphantom{+++++++}\vdots \\ a_{m1}x_1 + \cdots + a_{mn}x_n &= b_n\end{align*}\right. \qquad \text{ where } a_{ij}, b_{i} \in F

vault/Mathematics/Algebra/Linear Algebra/Vector Spaces/Bases/Basis Criteria.md

@@ -1,7 +1,7 @@
 >[!THEOREM] Theorem
 >Let $(V,K,+,\cdot)$ be a [finitely generated](../Spanning%20Set%20(Generator).md) [vector space](../Vector%20Space.md).
 >
->Every [set](../../../../Set%20Theory/index.md) $B$ of $\dim(V)$ [linearly independent](../Linear%20Independence.md) [vectors](../Vector%20Space.md) forms a [basis](Basis.md) of $(V,K,+,\cdot)$.
+>Every [set](../../../../Set%20Theory/Sets.md) $B$ of $\dim(V)$ [linearly independent](../Linear%20Independence.md) [vectors](../Vector%20Space.md) forms a [basis](Basis.md) of $(V,K,+,\cdot)$.
 >
 >>[!PROOF]-
 >>

vault/Mathematics/Algebra/Linear Algebra/Vector Spaces/Bases/Basis.md

@@ -4,7 +4,7 @@
 >
 >>[!THEOREM]- Equivalent Definition
 >>
->>A [set](../../../../Set%20Theory/index.md) $B$ of [vectors](../Vector%20Space.md) is a [basis](Basis.md) of a [vector space](../Vector%20Space.md) $(V,F,+,\cdot)$ if and only if $B$ is a [maximally linearly independent](../Linear%20Independence.md).
+>>A [set](../../../../Set%20Theory/Sets.md) $B$ of [vectors](../Vector%20Space.md) is a [basis](Basis.md) of a [vector space](../Vector%20Space.md) $(V,F,+,\cdot)$ if and only if $B$ is a [maximally linearly independent](../Linear%20Independence.md).
 >>
 >>>[!PROOF]-
 >>>
@@ -14,7 +14,7 @@
 >
 >>[!THEOREM]- Equivalent Definition
 >>
->>A [set](../../../../Set%20Theory/index.md) $B$ of [vectors](../Vector%20Space.md) from is a [basis](Basis.md) of a [vector space](../Vector%20Space.md) $(V,F,+,\cdot)$ if and only if $B$ is a [minimal spanning set](../Spanning%20Set%20(Generator).md).
+>>A [set](../../../../Set%20Theory/Sets.md) $B$ of [vectors](../Vector%20Space.md) from is a [basis](Basis.md) of a [vector space](../Vector%20Space.md) $(V,F,+,\cdot)$ if and only if $B$ is a [minimal spanning set](../Spanning%20Set%20(Generator).md).
 >>
 >>>[!PROOF]-
 >>>

vault/Mathematics/Algebra/Linear Algebra/Vector Spaces/Bases/Steinitz Exchange Lemma.md

@@ -2,7 +2,7 @@
 >
 >Let $B = \{\mathbf{v}_1,\cdots,\mathbf{v}_n\}$ be a [basis](Basis.md) of a [finitely generated](../Spanning%20Set%20(Generator).md) [vector space](../Vector%20Space.md) $(V, K, +,\cdot)$.
 >
->For every [set](../../../../Set%20Theory/index.md) $\{\mathbf{u}_1,\cdots,\mathbf{u}_m\}\subset V$ of [linearly independent](../Linear%20Independence.md) [vectors](../Vector%20Space.md)  there are $n-m$ [vectors](../Vector%20Space.md) in $B$ (without loss of generality - $\mathbf{v}_{m+1}, \cdots, \mathbf{v}_n$) such that
+>For every [set](../../../../Set%20Theory/Sets.md) $\{\mathbf{u}_1,\cdots,\mathbf{u}_m\}\subset V$ of [linearly independent](../Linear%20Independence.md) [vectors](../Vector%20Space.md)  there are $n-m$ [vectors](../Vector%20Space.md) in $B$ (without loss of generality - $\mathbf{v}_{m+1}, \cdots, \mathbf{v}_n$) such that
 >
 >$$
 >\{\mathbf{u}_1,\cdots,\mathbf{u}_m,\mathbf{v}_{m+1},\cdots,\mathbf{v}_n\}

vault/Mathematics/Algebra/Linear Algebra/Vector Spaces/Inner Product Spaces/Orthogonal Complement.md

@@ -2,7 +2,7 @@
 >
 >Let $(U,F,+,\cdot)$ be a [subspace](../Subspace.md) of an [inner product space](Inner%20Product%20Space.md) $(V,F,+,\cdot)$.
 >
->The **orthogonal complement** of $U$ is the [set](../../../../Set%20Theory/index.md) of [vectors](../Vector%20Space.md) in $V$ which are [orthogonal](Orthogonality.md) to all [vectors](../Vector%20Space.md) in $U$.
+>The **orthogonal complement** of $U$ is the [set](../../../../Set%20Theory/Sets.md) of [vectors](../Vector%20Space.md) in $V$ which are [orthogonal](Orthogonality.md) to all [vectors](../Vector%20Space.md) in $U$.
 >
 >$$
 >\{\mathbf{v}\in V \mid \mathbf{v}\perp\mathbf{u}, \,\,\, \forall \mathbf{u}\in U\}

vault/Mathematics/Algebra/Linear Algebra/Vector Spaces/Linear Independence.md

@@ -10,21 +10,21 @@
 >
 >>[!DEFINITION] Definition: Maximality
 >>
->>A [set](../../../Set%20Theory/index.md) of [linearly independent](Linear%20Independence.md) [vectors](Vector.md) $S = \{ \mathbf{v}_1, \mathbf{v}_2, \cdots \}$ is **maximal** if there is no $\mathbf{v} \in V \setminus S$ such that $S \cup \{ \mathbf{v} \}$ are still [linearly independent](Linear%20Independence.md). 
+>>A [set](../../../Set%20Theory/Sets.md) of [linearly independent](Linear%20Independence.md) [vectors](Vector.md) $S = \{ \mathbf{v}_1, \mathbf{v}_2, \cdots \}$ is **maximal** if there is no $\mathbf{v} \in V \setminus S$ such that $S \cup \{ \mathbf{v} \}$ are still [linearly independent](Linear%20Independence.md). 
 >>
 >
 
 >[!THEOREM] Theorem: Size Limit for Linearly Independent Sets
 >
->The number of elements in any [set](../../../Set%20Theory/index.md) $I$ of [linearly independent](Linear%20Independence.md) [vectors](Vector.md) from a [finitely generated](Vector%20Spaces/Spanning%20Set%20(Generator).md) [vector space](Vector%20Space.md) $(V,F,+,\cdot)$ is always less than or equal to the [dimension](Bases/Dimension.md) of $V$.
+>The number of elements in any [set](../../../Set%20Theory/Sets.md) $I$ of [linearly independent](Linear%20Independence.md) [vectors](Vector.md) from a [finitely generated](Vector%20Spaces/Spanning%20Set%20(Generator).md) [vector space](Vector%20Space.md) $(V,F,+,\cdot)$ is always less than or equal to the [dimension](Bases/Dimension.md) of $V$.
 >
 >$$
 >|I| \le \dim(V)
 >$$
 >
 >>[!PROOF]-
 >>
->>Let $B = \{\mathbf{b}_1, \cdots, \mathbf{b}_n,\}$ be a [basis](Bases/Basis.md) of $V$ and let $I = \{\mathbf{v}_1, \cdots, \mathbf{v}_m\}$ be a [set](../../../Set%20Theory/index.md) of [linearly independent](Linear%20Independence.md) [vectors](Vector%20Space.md).
+>>Let $B = \{\mathbf{b}_1, \cdots, \mathbf{b}_n,\}$ be a [basis](Bases/Basis.md) of $V$ and let $I = \{\mathbf{v}_1, \cdots, \mathbf{v}_m\}$ be a [set](../../../Set%20Theory/Sets.md) of [linearly independent](Linear%20Independence.md) [vectors](Vector%20Space.md).
 >>
 >>According to the [Steinitz exchange lemma](Bases/Steinitz%20Exchange%20Lemma.md) there are $n-m$ [vectors](Vector%20Space.md) in $B$ which form a basis with the vectors from $I$. This means that $n-m$ cannot be negative and thus the proof is complete.
 >>

vault/Mathematics/Algebra/Linear Algebra/Vector Spaces/Span.md

@@ -1,6 +1,6 @@
 >[!DEFINITION] Definition: Span
 >
->The **span** of a [set](../../../Set%20Theory/index.md) of [vectors](Vector%20Space.md) $\{\mathbf{v}_1, \mathbf{v}_1, \cdots \}$ in some [vector space](Vector%20Space.md) $(V,K,+,\cdot)$ is the [set](../../../Set%20Theory/index.md) of all [linear combinations](Linear%20Combination.md) which can be constructed from them.
+>The **span** of a [set](../../../Set%20Theory/Sets.md) of [vectors](Vector%20Space.md) $\{\mathbf{v}_1, \mathbf{v}_1, \cdots \}$ in some [vector space](Vector%20Space.md) $(V,K,+,\cdot)$ is the [set](../../../Set%20Theory/Sets.md) of all [linear combinations](Linear%20Combination.md) which can be constructed from them.
 >
 >$$
 >\{ c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots \mid c_i \in F\}

vault/Mathematics/Algebra/Linear Algebra/Vector Spaces/Spanning Set (Generator).md

@@ -2,7 +2,7 @@
 >
 >Let $(V,F,+,\cdot)$ be a [vector space](Vector%20Space.md).
 >
->A [set](../../../Set%20Theory/index.md) of [vectors](Vector%20Space.md) $S = \{\mathbf{v}_1, \mathbf{v}_2, \cdots \}$ is called a **spanning set** or a **generator** of $V$ if their [span](Span.md) is equal to $V$, i.e. every vector in $V$ can be expressed as a linear combination of vectors from $\{\mathbf{v}_1, \mathbf{v}_2, \cdots \}$:
+>A [set](../../../Set%20Theory/Sets.md) of [vectors](Vector%20Space.md) $S = \{\mathbf{v}_1, \mathbf{v}_2, \cdots \}$ is called a **spanning set** or a **generator** of $V$ if their [span](Span.md) is equal to $V$, i.e. every vector in $V$ can be expressed as a linear combination of vectors from $\{\mathbf{v}_1, \mathbf{v}_2, \cdots \}$:
 >
 >$$
 >\operatorname{span}(\mathbf{v}_1, \mathbf{v}_2, \cdots) = V

vault/Mathematics/Algebra/Linear Algebra/Vector Spaces/Vector Space.md

@@ -1,6 +1,6 @@
 >[!DEFINITION] Definition: Vector Space
 >
->A **vector space** $(V,F,+,\cdot)$ consists of a non-[empty](../../../Set%20Theory/The%20Empty%20Set.md) [set](../../../Set%20Theory/index.md) $V$ and a [field](../../Fields/index.md) $F$ which are equipped with two [operations](../../../Analysis/Functions/index.md) - a **vector addition** $+: V \times V \to V$ and a **scalar multiplication** $\cdot: V \times F \to V$ - which have the following properties:
+>A **vector space** $(V,F,+,\cdot)$ consists of a non-[empty](../../../Set%20Theory/The%20Empty%20Set.md) [set](../../../Set%20Theory/Sets.md) $V$ and a [field](../../Fields/index.md) $F$ which are equipped with two [operations](../../../Analysis/Functions/index.md) - a **vector addition** $+: V \times V \to V$ and a **scalar multiplication** $\cdot: V \times F \to V$ - which have the following properties:
 >
 >- Commutativity: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \qquad \forall \mathbf{u},\mathbf{v} \in V$
 >- Associativity I: $\mathbf{u} + (\mathbf{v} + \mathbf{w}) = (\mathbf{u} + \mathbf{v}) + \mathbf{w} \qquad \forall \mathbf{u},\mathbf{v}, \mathbf{w} \in V$

vault/Mathematics/Algebra/Monoid.md

@@ -9,7 +9,7 @@ tags:
 
 >[!DEFINITION] Definition: Monoid
 >
->A **monoid** $(M, \circ)$ is a [set](../Set%20Theory/index.md) $M$ equipped with a binary [operation](../Analysis/Functions/index.md) $\circ: M \times M \to M$ which has the following properties.
+>A **monoid** $(M, \circ)$ is a [set](../Set%20Theory/Sets.md) $M$ equipped with a binary [operation](../Analysis/Functions/index.md) $\circ: M \times M \to M$ which has the following properties.
 >- Associativity: $(a \circ b) \cdot c = a \circ (b \circ c)$
 >- Existence of an identity element: $\exists e \in M: a \circ e = e \circ a = a$
 >