Elementary Results about Functions and Sets

This note aggregates many elementary results about images and pre-images of sets under functions, and how these properties differ when imposing conditions such as injectivity, surjectivity and bijectivity.

Image and Pre-Image of Union and Intersection

Image Preserves Union

For \(f : X \to Y\) and \(A_i \subseteq X\) for all \(i \in I\)

\[ f\left(\bigcup_{i \in I} A_{i}\right) = \bigcup_{i \in I} f(A_{i})\]

Image of Intersection is Subset of Intersection of Images

For \(f : X \to Y\) and \(A_i \subseteq X\) for all \(i \in I\)

\[ f\left(\bigcap_{i \in I} A_{i}\right) \subseteq \bigcap_{i \in I} f(A_{i})\]

and in particular

\[ f\left(\bigcap_{i \in I} A_{i}\right) = \bigcap_{i \in I} f(A_{i}) \quad \text{for any collection of}\ A_i \iff f \ \text{is injective} \]

Pre-Image Preserves Union

For \(f : X \to Y\) and \(B_i \subseteq Y\) for all \(i \in I\)

\[ f^{-1}\left(\bigcup_{i \in I} B_{i}\right) = \bigcup_{i \in I} f^{-1}(B_{i})\]

Pre-Image Preserves Intersection

For \(f : X \to Y\) and \(B_i \subseteq Y\) for all \(i \in I\)

\[ f^{-1}\left(\bigcap_{i \in I} B_{i}\right) = \bigcap_{i \in I} f^{-1}(B_{i})\]

Preservation of Inclusions

Images Preserve Inclusions

For \(f : X \to Y\) and \(A_1, A_2 \subseteq X\)

\[ A_{1} \subseteq A_{2} \implies f(A_{1}) \subseteq f(A_{2})\]

Pre-Images Preserve Inclusions

For \(f : X \to Y\) and \(B_1, B_2 \subseteq Y\)

\[ B_1 \subseteq B_2 \implies f^{-1}(B_1) \subseteq f^{-1}(B_2)\]

Image of Pre-Image and Pre-Image of Image

Image of Pre-Image is Subset of Set

For \(f : X \to Y\) and \(B \subseteq Y\)

\[ f(f^{-1}(B)) \subseteq B\]

and in particular

\[ f(f^{-1}(B)) = B \quad \forall B \subseteq Y \iff f \ \text{is surjective}\]

Set is Subset of Pre-Image of Image

For \(f : X \to Y\) and \(A \subseteq X\)

\[ A \subseteq f^{-1}(f(A))\]

and in particular

\[ A = f^{-1}(f(A)) \quad \forall A \subseteq X \iff f \ \text{is injective}\]

Image and Pre-Image of Set Complements

Pre-Image Preserves Complement

For \(f : X \to Y\) and \(B \subseteq Y\)

\[ f^{-1}(B^{c}) = f^{-1}(B)^{c}\]

Image of Set Complement

For \(f : X \to Y\)

\[ f(A)^{c} \subseteq f(A^{c}) \quad \forall A \subseteq X \iff f \ \text{is surjective}\]

with the equality condition

\[ f(A)^{c} = f(A^{c}) \quad \forall A \subseteq X \iff f \ \text{is bijective}\]

Set Difference

Pre-Image Preserves Set Difference

For \(f : X \to Y\) and \(B_1, B_2 \subseteq Y\)

\[ f^{-1}(B_{1} - B_{2}) = f^{-1}(B_{1}) - f^{-1}(B_{2})\]