Editing Image (mathematics) (section)

=== General ===

For every function <math>f : X \to Y</math> and all subsets <math>A \subseteq X</math> and <math>B \subseteq Y,</math> the following properties hold:

{| class="wikitable"
|-
! Image
! Preimage
|-
|<math>f(X) \subseteq Y</math>
|<math>f^{-1}(Y) = X</math>
|-
|<math>f\left(f^{-1}(Y)\right) = f(X)</math>
|<math>f^{-1}(f(X)) = X</math>
|-
|<math>f\left(f^{-1}(B)\right) \subseteq B</math><br>(equal if <math>B \subseteq f(X);</math> for instance, if <math>f</math> is surjective)<ref name="halmos-1960-p31">See {{harvnb|Halmos|1960|p=31}}</ref><ref name="munkres-2000-p19">See {{harvnb|Munkres|2000|p=19}}</ref>
|<math>f^{-1}(f(A)) \supseteq A</math><br>(equal if <math>f</math> is injective)<ref name="halmos-1960-p31"/><ref name="munkres-2000-p19" />
|-
|<math>f(f^{-1}(B)) = B \cap f(X)</math>
|<math>\left(f \vert_A\right)^{-1}(B) = A \cap f^{-1}(B)</math>
|-
|<math>f\left(f^{-1}(f(A))\right) = f(A)</math>
|<math>f^{-1}\left(f\left(f^{-1}(B)\right)\right) = f^{-1}(B)</math>
|-
|<math>f(A) = \varnothing \,\text{ if and only if }\, A = \varnothing</math>
|<math>f^{-1}(B) = \varnothing \,\text{ if and only if }\, B \subseteq Y \setminus f(X)</math>
|-
|<math>f(A) \supseteq B \,\text{ if and only if } \text{ there exists } C \subseteq A \text{ such that } f(C) = B</math>
|<math>f^{-1}(B) \supseteq A \,\text{ if and only if }\, f(A) \subseteq B</math>
|-
|<math>f(A) \supseteq f(X \setminus A) \,\text{ if and only if }\, f(A) = f(X)</math>
|<math>f^{-1}(B) \supseteq f^{-1}(Y \setminus B) \,\text{ if and only if }\, f^{-1}(B) = X</math>
|-
|<math>f(X \setminus A) \supseteq f(X) \setminus f(A)</math>
|<math>f^{-1}(Y \setminus B) = X \setminus f^{-1}(B)</math><ref name="halmos-1960-p31" />
|-
|<math>f\left(A \cup f^{-1}(B)\right) \subseteq f(A) \cup B</math><ref name="lee-2010-p388">See p.388 of Lee, John M. (2010). Introduction to Topological Manifolds, 2nd Ed.</ref>
|<math>f^{-1}(f(A) \cup B) \supseteq A \cup f^{-1}(B)</math><ref name="lee-2010-p388" />
|-
|<math>f\left(A \cap f^{-1}(B)\right) = f(A) \cap B</math><ref name="lee-2010-p388" />
|<math>f^{-1}(f(A) \cap B) \supseteq A \cap f^{-1}(B)</math><ref name="lee-2010-p388" />
|}

Also:

* <math>f(A) \cap B = \varnothing \,\text{ if and only if }\, A \cap f^{-1}(B) = \varnothing</math>