preimage of a subset under a function

原像

Foundations & Logic · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: inverse image

Translation evidence: Literature usage

Meaning in context

For f:A→B and T⊆B, the members a of A such that f(a) belongs to T.

对于 f:A→B 和 T⊆B，由满足 f(a)∈T 的所有 a∈A 组成的集合。

Example

For f(n)=n2f(n)=n^2 on Z\mathbb Z, f−1[{4}]={−2,2}f^{-1}[\{4\}]=\{-2,2\}.

Usage and distinctions

Inverse-image notation does not assert that f has an inverse function.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Avigad, Hua, Lewis and van Doorn — Logic and Proof§15.1 The Function Concept; §15.2 Injective, Surjective, and Bijective Functions; §15.3 Functions and Subsets of the Domain · Checked 2026-10-14

Translation sources

  1. 鲍志强（北京大学）— 拓扑学2007年秋季教学纲要第1周9月12日，主要概念：「原像」 · Checked 2026-10-14
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