Images and Image Factorisation

Image of a Function

Definition

For $f : A \to B$, the image (or range) is:

$\text{Im}(f) = \{b \in B : \exists a \in A. f(a) = b\} = \{f(a) : a \in A\}$

Image as a Surjection

Define $\tilde{f} : A \twoheadrightarrow \text{Im}(f)$ by $\tilde{f}(a) = f(a)$.

This is surjective: for every $b \in \text{Im}(f)$, there exists $a$ with $f(a) = b$.

Image Factorisation

Theorem

Every function $f : A \to B$ factors uniquely as:

$f = m_f \circ \tilde{f}$

where:

• $\tilde{f} : A \twoheadrightarrow \text{Im}(f)$ is surjective
• $m_f : \text{Im}(f) \hookrightarrow B$ is injective (the inclusion)

Verification

For $a \in A$: $m_f(\tilde{f}(a)) = m_f(f(a)) = f(a)$ (since $m_f$ is inclusion).

Uniqueness

If $f = m \circ s$ with $s$ surjective and $m$ injective, then $\text{Im}(s) = \text{Im}(f)$ and $m = m_f$ up to isomorphism.

Relational Images

Direct Image

For a relation $R : A \nrightarrow B$ and $X \subseteq A$:

$R_*(X) = \{b \in B : \exists x \in X. xRb\}$

This is the direct image of $X$ under $R$.

Inverse Image

For $Y \subseteq B$:

$R^*(Y) = \{a \in A : \forall b. (aRb \Rightarrow b \in Y)\}$

This is the inverse image (or preimage) under $R$.

For Functions

For $f : A \to B$:

• $f_*(X) = \{f(x) : x \in X\}$ (forward image)
• $f^*(Y) = f^{-1}(Y) = \{a : f(a) \in Y\}$ (inverse image)

Properties of Inverse Images

Preservation of Set Operations

For $f : A \to B$:

$f^{-1}(Y_1 \cap Y_2) = f^{-1}(Y_1) \cap f^{-1}(Y_2)$ $f^{-1}(Y_1 \cup Y_2) = f^{-1}(Y_1) \cup f^{-1}(Y_2)$ $f^{-1}(Y^c) = (f^{-1}(Y))^c$ $f^{-1}(\emptyset) = \emptyset$ $f^{-1}(B) = A$

Galois Connection

There is a Galois connection between $f_*$ and $f^*$:

$f_*(X) \subseteq Y \iff X \subseteq f^*(Y)$

Applications

Surjectivity Characterisation

$f$ is surjective iff $\text{Im}(f) = B$ iff $f_*(A) = B$.

Injectivity Characterisation

$f$ is injective iff $f^*(f_*(X)) = X$ for all $X \subseteq A$.

Bijection Characterisation

$f$ is bijective iff $f_*$ and $f^*$ are mutually inverse bijections between $\mathcal{P}(A)$ and $\mathcal{P}(B)$.

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Summary

• $\text{Im}(f) = \{f(a) : a \in A\}$
• Every $f : A \to B$ factors as $A \twoheadrightarrow \text{Im}(f) \hookrightarrow B$
• Direct image $f_*(X) = \{f(x) : x \in X\}$
• Inverse image $f^*(Y) = f^{-1}(Y) = \{a : f(a) \in Y\}$
• Inverse images preserve all Boolean operations