User:Ascii/ProofWiki Sampling Notes for Theorems/Mapping Theory


 * 1) Image of Subset under Mapping is Subset of Image
 * Let $f: S \to T$ be a mapping from $S$ to $T$ and $A, B \subseteq S$ such that $A \subseteq B$.
 * Then $A \subseteq B \implies f \sqbrk A \subseteq f \sqbrk B$
 * 1) Inverse of Mapping is One-to-Many Relation
 * Let $f$ be a mapping.
 * Then its inverse $f^{-1}$ is a one-to-many relation.
 * 1) Preimage of Mapping equals Domain
 * The preimage of a mapping is the same set as its domain: $\Preimg f = \Dom f$
 * 1) Null Relation is Mapping iff Domain is Empty Set
 * Let $S$ and $T$ be sets.
 * The null relation $\mathcal R = \varnothing \subseteq S \times T$ is a mapping iff $S = \varnothing$.
 * 1) Mapping is Constant iff Image is Singleton
 * A mapping is a constant mapping its image is a singleton.
 * 1) Identity Mapping is Left Identity
 * Let $f: S \to T$ be a mapping.
 * Then $I_T \circ f = f$
 * 1) Identity Mapping is Right Identity
 * Let $f: S \to T$ be a mapping.
 * Then $f \circ I_S = f$

Injections

 * 1) Equivalence of Definitions of Injection
 * $\forall x_1, x_2 \in \Dom f: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$
 * $\forall x_1, x_2 \in \Dom f: x_1 \ne x_2 \implies \map f {x_1} \ne \map f {x_2}$
 * An injection is a relation which is both one-to-one and left-total.
 * $f^{-1} {\restriction_{\Img f} }: \Img f \to \Dom f$ is a mapping
 * $\forall y \in \Img f: \card {\map {f^{-1} } y} = \card {\set {f^{-1} \sqbrk {\set y} } } = 1$
 * $\exists g: T \to S: g \circ f = I_S$
 * $f$ is an injection $f$ is left cancellable
 * 1) Injection iff Left Cancellable
 * A mapping $f$ is an injection $f$ is left cancellable.
 * 1) Identity Mapping is Injection
 * On any set $S$, the identity mapping $I_S: S \to S$ is an injection.
 * 1) Composite of Injections is Injection
 * #: If $g$ and $f$ are injections, then so is the composite $g \circ f$.
 * 1) Injection if Composite is Injection
 * Let $f$ and $g$ be mappings such that their composite $g \circ f$ is an injection.
 * Then $f$ is an injection.
 * 1) Injection iff Left Inverse
 * A mapping $f: S \to T, S \ne \O$ is an injection : $\exists g: T \to S: g \circ f = I_S$
 * That is, $f$ has a left inverse.
 * 1) Inclusion Mapping is Injection
 * The inclusion mapping $i_S: S \to T$ defined as: $\forall x \in S: i_S \left({x}\right) = x$ is an injection

Surjections

 * 1) Equivalence of Definitions of Surjection
 * $f: S \to T$ is a surjection $\forall y \in T: \exists x \in \Dom f: \map f x = y$ (that is,  $f$ is right-total).
 * $f: S \to T$ is a surjection : $f \sqbrk S = T$ (that is, $f$ is a surjection its image equals its codomain: $\Img f = \Cdm f$)
 * 1) Surjection iff Right Cancellable
 * A mapping $f$ is a surjection $f$ is right cancellable.
 * 1) Identity Mapping is Surjection
 * On any set $S$, the identity mapping $I_S: S \to S$ is a surjection.
 * 1) Composite of Surjections is Surjection
 * If $g$ and $f$ are surjections, then so is the composite $g \circ f$.
 * 1) Surjection iff Right Inverse
 * A mapping $f: S \to T, S \ne \O$ is a surjection : $\exists g: T \to S: f \circ g = I_T$
 * That is, $f$ has a right inverse.
 * 1) Surjection if Composite is Surjection
 * Let $f: S_1 \to S_2$ and $g: S_2 \to S_3$ be mappings such that $g \circ f$ is a surjection.
 * Then $g$ is a surjection.

Bijections

 * 1) Identity Mapping is Bijection
 * The identity mapping $I_S: S \to S$ on the set $S$ is a bijection.
 * 1) Identity Mapping is Permutation
 * The identity mapping $I_S: S \to S$ on the set $S$ is a permutation.
 * 1) Equivalence of Definitions of Bijection
 * $f$ is both an injection and a surjection.
 * $f$ has both a left inverse and a right inverse.
 * The inverse $f^{-1}$ of $f$ is a mapping from $T$ to $S$.
 * For each $y \in T$ there exists one and only one $x \in S$ such that $\tuple {x, y} \in f$.
 * For each $x \in S$ there exists one and only one $y \in T$ such that $\left({x, y}\right) \in f$ and for each $y \in T$ there exists one and only one $x \in S$ such that $\left({x, y}\right) \in f$.
 * 1) Composite of Bijection with Inverse is Identity Mapping
 * Let $f: S \to T$ be a bijection.
 * Then $f^{-1} \circ f = I_S$ and $f \circ f^{-1} = I_T$
 * 1) Inverse of Inverse of Bijection
 * Let $f: S \to T$ be a bijection.
 * Then $\paren {f^{-1} }^{-1} = f$
 * 1) Composite of Bijections is Bijection
 * If $f$ and $g$ are both bijections, then so is the composite $f \circ g$.
 * 1) Inverse of Composite Bijection
 * Let $f$ and $g$ be bijections.
 * Then $\left({g \circ f}\right)^{-1} = f^{-1} \circ g^{-1}$ and $f^{-1} \circ g^{-1}$ is itself a bijection.
 * 1) Composite of Permutations is Permutation
 * Let $f, g$ are permutations of a set $S$.
 * Then their composite $g \circ f$ is also a permutation of $S$.
 * 1) Inverse of Permutation is Permutation
 * If $f$ is a permutation of $S$, then so is its inverse $f^{-1}$.
 * 1) Left Inverse Mapping is Surjection
 * Let $f: S \to T$ be an injection and $g: T \to S$ be a left inverse of $f$.
 * Then $g$ is a surjection.
 * 1) Right Inverse Mapping is Injection
 * Let $f: S \to T$ be a mapping and $g: T \to S$ be a right inverse of $f$.
 * Then $g$ is an injection.
 * 1) Restriction of Mapping to Image is Surjection
 * Let $f: S \to T$ be a mapping and $g: S \to \Img f$ be the restriction of $f$ to $S \times \Img f$.
 * Then $g$ is a surjective restriction of $f$.

Images

 * 1) Image of Union under Mapping
 * Let $S$ and $T$ be sets, $f: S \to T$ be a mapping, and $A$ and $B$ be subsets of $S$.
 * Then $f \sqbrk {A \cup B} = f \sqbrk A \cup f \sqbrk B$
 * 1) Image of Intersection under Mapping
 * Let $S$ and $T$ be sets, $f: S \to T$ be a mapping, and $A$ and $B$ be subsets of $S$.
 * Then $f \sqbrk {A \cap B} \subseteq f \sqbrk A \cup f \sqbrk B$
 * 1) Image of Intersection under Injection
 * Let $S$ and $T$ be sets, $f: S \to T$ be a mapping, and $A$ and $B$ be subsets of $S$.
 * Then $f \sqbrk {A \cap B} = f \sqbrk A \cup f \sqbrk B$ $f$ is an injection.
 * 1) Preimage of Union under Mapping
 * Let $S$ and $T$ be sets, $f: S \to T$ be a mapping, and $T_1$ and $T_2$ be subsets of $T$.
 * Then $f^{-1} \sqbrk {T_1 \cup T_2} = f^{-1} \sqbrk {T_1} \cup f^{-1} \sqbrk {T_2}$
 * 1) Preimage of Intersection under Mapping
 * Let $S$ and $T$ be sets, $f: S \to T$ be a mapping, and $T_1$ and $T_2$ be subsets of $T$.
 * Then $f^{-1} \sqbrk {T_1 \cap T_2} = f^{-1} \sqbrk {T_1} \cap f^{-1} \sqbrk {T_2}$
 * 1) Image of Set Difference under Mapping
 * Let $f: S \to T$ be a mapping and $S_1$ and $S_2$ be subsets of $S$.
 * Then $f \sqbrk {S_1} \setminus f \sqbrk {S_2} \subseteq f \sqbrk {S_1 \setminus S_2}$
 * 1) Image of Set Difference under Injection
 * Let $f: S \to T$ be a mapping and $S_1$ and $S_2$ be subsets of $S$.
 * Then $\forall S_1, S_2 \subseteq S: f \left[{S_1}\right] \setminus f \left[{S_2}\right] = f \left[{S_1 \setminus S_2}\right]$
 * 1) Preimage of Set Difference under Mapping
 * Let $f: S \to T$ be a mapping and $T_1$ and $T_2$ be subsets of $T$.
 * Then $f^{-1} \sqbrk {T_1 \setminus T_2} = f^{-1} \sqbrk {T_1} \setminus f^{-1} \sqbrk {T_2}$
 * 1) Quotient Mapping is Surjection
 * Let $\mathcal R$ be an equivalence relation on $S$.
 * Then the quotient mapping $q_{\mathcal R}: S \to S / \mathcal R$ is a surjection.
 * 1) Induced Equivalence is Equivalence Relation
 * Let $f: S \to T$ be a mapping and $\mathcal R_f \subseteq S \times S$ be the relation induced by $f$.
 * Then $\mathcal R_f$ is an equivalence relation.