Image and preïmage

Given a function $𝑓 : 𝑋 \to 𝑌$ , the image¹ $𝑓 ⋆ (𝐴)$ of a subset $𝐴 \subseteq 𝑋$ is the set containing all output values $𝑓 (𝑎)$ for $𝑎 \in 𝐴$ #m/def/general, whereas the preïmage² $𝑓 ⋆ (𝐵)$ of a subset $𝐵 \subseteq 𝑌$ is the set containing all elements of $𝑋$ with an image in $𝐵$ . #m/def/general

𝑓 ⋆ (𝐴) = {𝑓 (𝑎) : 𝑎 \in 𝐴} 𝑓 ⋆ (𝐵) = {𝑥 \in 𝑋 : 𝑓 (𝑥) \in 𝐵}

Properties

A stack exchange answer

The preïmage of the image and image of the preïmage are not necessarily the identity
The image map of a bijection is a bijection
$𝑔 ⋆ 𝑓 ⋆ = (𝑓 \circ 𝑔) ⋆$
$𝑓 ⋆ 𝑔 ⋆ = (𝑓 \circ 𝑔) ⋆$

#state/tidy | #lang/en | #SemBr

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Image and preïmage

Properties

Footnotes