It looks like you're new here. If you want to get involved, click one of these buttons!

- All Categories 2.2K
- Applied Category Theory Course 352
- Applied Category Theory Seminar 4
- Exercises 149
- Discussion Groups 49
- How to Use MathJax 15
- Chat 479
- Azimuth Code Project 108
- News and Information 145
- Azimuth Blog 149
- Azimuth Forum 29
- Azimuth Project 189
- - Strategy 108
- - Conventions and Policies 21
- - Questions 43
- Azimuth Wiki 711
- - Latest Changes 701
- - - Action 14
- - - Biodiversity 8
- - - Books 2
- - - Carbon 9
- - - Computational methods 38
- - - Climate 53
- - - Earth science 23
- - - Ecology 43
- - - Energy 29
- - - Experiments 30
- - - Geoengineering 0
- - - Mathematical methods 69
- - - Meta 9
- - - Methodology 16
- - - Natural resources 7
- - - Oceans 4
- - - Organizations 34
- - - People 6
- - - Publishing 4
- - - Reports 3
- - - Software 21
- - - Statistical methods 2
- - - Sustainability 4
- - - Things to do 2
- - - Visualisation 1
- General 39

Options

Check the two claims made in Proposition 1.53.

**Proposition 1.53**. For any preorder \( (P, \le_P ) \), the identity function is monotone.

If \( (Q, \le_Q ) \) and \( (R, \le_R) \) are preorders and \( f : P \rightarrow Q \) and \( g : Q \rightarrow R \) are monotone, then \( ( f .g) : P \rightarrow R \) is also monotone.

## Comments

Let \(x_p\) and \(y_p\) be elements of \(P\) such that \(x_p\leq y_p\).

Since \(f\) is monotone, it maps to elements \(x_q\) and \(y_q\), where \(x_q\leq y_q\)

Monotone function \(g\) maps \(x_q\) and \(y_q\) to \(x_r\) and \(y_r\) such that \(x_r\leq y_r\)

Thus, \( ( f .g) : P \rightarrow R \) maps \(x_p\) and \(y_p\) to \(x_r\) and \(y_r\), since \(x_p\leq y_p\) and \(x_r\leq y_r\), \( ( f .g) \) is monotone

`Let \\(x_p\\) and \\(y_p\\) be elements of \\(P\\) such that \\(x_p\leq y_p\\). Since \\(f\\) is monotone, it maps to elements \\(x_q\\) and \\(y_q\\), where \\(x_q\leq y_q\\) Monotone function \\(g\\) maps \\(x_q\\) and \\(y_q\\) to \\(x_r\\) and \\(y_r\\) such that \\(x_r\leq y_r\\) Thus, \\( ( f .g) : P \rightarrow R \\) maps \\(x_p\\) and \\(y_p\\) to \\(x_r\\) and \\(y_r\\), since \\(x_p\leq y_p\\) and \\(x_r\leq y_r\\), \\( ( f .g) \\) is monotone`

Looks good, Deepak! Like 99.5% of mathematicians, I usually use the notation \(g \circ f\) or just \(gf\) for the composite of functions \(f : P \to Q\) and \(g : Q \to R\), but Fong and Spivak use \( (f.g) \), which has certain advantages.

`Looks good, Deepak! Like 99.5% of mathematicians, I usually use the notation \\(g \circ f\\) or just \\(gf\\) for the composite of functions \\(f : P \to Q\\) and \\(g : Q \to R\\), but Fong and Spivak use \\( (f.g) \\), which has certain advantages.`