You are here
Homeproperties of functions
Primary tabs
properties of functions
Let $f\colon X\to Y$ be a function. Let $(A_{i})_{{i\in I}}$ be a family of subsets of $X$, and let $(B_{j})_{{j\in J}}$ be a family of subsets of $Y$, where $I$ and $J$ are nonempty index sets.
Then, it is easy to prove, directly from definitions, that the following hold:

$f(\bigcap\limits_{{i\in I}}{A_{i}})\subseteq\bigcap\limits_{{i\in I}}{f(A_{i})}$ (i.e., the image of an intersection is contained in the intersection of the images)

$A\subseteq f^{{1}}(f(A))$ for any $A\subseteq X$ (where $f^{{1}}(f(A))$ is the inverse image of $f(A)$)

$f(f^{{1}}(B))\subseteq B$ for any $B\subseteq Y$

$f^{{1}}(Y\setminus B)=X\setminus f^{{1}}(B)$ for any $B\subseteq Y$

$f^{{1}}(\bigcup\limits_{{j\in J}}{B_{j}})=\bigcup\limits_{{j\in J}}{f^{{1}}(% B_{j})}$ (the inverse image of a union is the union of the inverse images)

$f^{{1}}(\bigcap\limits_{{j\in J}}{B_{j}})=\bigcap\limits_{{j\in J}}{f^{{1}}(% B_{j})}$ (the inverse image of an intersection is the intersection of the inverse images)

$f(f^{{1}}(B))=B$ for every $B\subseteq Y$ if and only if $f$ is surjective.
For more properties related specifically to inverse images, see the inverse image entry.
Further, the following conditions are equivalent (for more, see the entry on injective functions):

$f$ is injective

$f(S\cap T)=f(S)\cap f(T)$ for all $S,T\subseteq X$

$f^{{1}}(f(S))=S$ for all $S\subseteq X$

$f(S)\cap f(T)=\varnothing$ for all $S,T\subseteq X$ such that $S\cap T=\varnothing$

$f(S\setminus T)=f(S)\setminus f(T)$ for all $S,T\subseteq X$
Mathematics Subject Classification
03E20 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new correction: Error in proof of Proposition 2 by alex2907
Jun 24
new question: A good question by Ron Castillo
Jun 23
new question: A trascendental number. by Ron Castillo
Jun 19
new question: Banach lattice valued Bochner integrals by math ias
Corrections
second property is false by yark ✓
second property by yark ✓
Typo? by ratboy ✓
missing cached output by CWoo ✓