404: Map Preimages of Disjoint Subsets Are Disjoint

<The previous article in this series | The table of contents of this series | The next article in this series>

A description/proof of that map preimages of disjoint subsets are disjoint

---

Topics

About: set

---

The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

---

Starting Context

• The reader knows a definition of set.
• The reader knows a definition of map.

---

Target Context

• The reader will have a description and a proof of the proposition that the preimages of any disjoint subsets under any map are disjoint.

---

Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

---

Main Body

---

1: Description

For any sets, $S_1,S_2$, any map, $f:S_1\to S_2$, and any disjoint subsets, $S_{21},S_{22}\subseteq S_2$, such that $S_{21}\cap S_{22}=\mathrm{\varnothing }$, $f^{-1}(S_{21})\cap f^{-1}(S_{22})=\mathrm{\varnothing }$.

---

2: Proof

Suppose that there was a common element, $p\in f^{-1}(S_{21})$ and $p\in f^{-1}(S_{22})$. $f(p)\in S_{21}$ and $f(p)\in S_{22}$, a contradiction.

---

References

<The previous article in this series | The table of contents of this series | The next article in this series>