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

$$f\left(\bigcap _{i\in I}{A}_i\right)=\bigcap _{i\in I}f(A_i)\text{for any collection of}{A}_i⟺f\text{is injective}$$

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 $A_1,A_2\subseteq X$

$$A_1\subseteq A_2⟹f(A_1)\subseteq f(A_2)$$

Pre-Images Preserve Inclusions

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

$$B_1\subseteq B_2⟹f^{-1}(B_1)\subseteq f^{-1}(B_2)$$

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

$$f(f^{-1}(B))=B\mathrm{\forall }B\subseteq Y⟺f\text{is surjective}$$

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

$$A=f^{-1}(f(A))\mathrm{\forall }A\subseteq X⟺f\text{is injective}$$

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 $f:X\to Y$

$$f(A{)}^c\subseteq f(A^c)\mathrm{\forall }A\subseteq X⟺f\text{is surjective}$$

with the equality condition

$$f(A{)}^c=f(A^c)\mathrm{\forall }A\subseteq X⟺f\text{is bijective}$$

Set Difference

Pre-Image Preserves Set Difference

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

$$f^{-1}(B_1-B_2)=f^{-1}(B_1)-f^{-1}(B_2)$$