Function

Image and preïmage

Given a function 𝑓 :𝑋 →𝑌, the image¹ 𝑓⋆(𝐴) of a subset 𝐴 ⊆𝑋 is the set containing all output values 𝑓(𝑎) for 𝑎 ∈𝐴 general, whereas the preïmage² 𝑓⋆(𝐵) of a subset 𝐵 ⊆𝑌 is the set containing all elements of 𝑋 with an image in 𝐵. general

𝑓⋆(𝐴)={𝑓(𝑎):𝑎∈𝐴}𝑓⋆(𝐵)={𝑥∈𝑋:𝑓(𝑥)∈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 ∘𝑔)⋆
• 𝑓⋆𝑔⋆ =(𝑓 ∘𝑔)⋆

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Footnotes

1. German Bild ↩

2. German Urbild ↩