Options

Exercise 37 - Chapter 4

Let \(+\colon\mathbb{R}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) be as in Example 4.36. What is its conjoint \(\check{+}\)?

Example 4.36:

Consider the function \(+\colon\mathbb{R}\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}\), sending a triple \((a,b,c)\) of real numbers to \(a+b+c\in\mathbb{R}\). This function is monotonic, because if \((a,b,c)\leq(a',b',c')\)---i.e. if \(a\leq a'\) and \(b\leq b'\), and \(c\leq c'\)---then obviously \(a+b+c\leq a'+b'+c'\). Thus it has a companion and a conjoint.

Its companion \(\hat{+}\colon(\mathbb{R}\times\mathbb{R}\times\mathbb{R})\nrightarrow\mathbb{R}\) is the function that sends \((a,b,c,d)\) to \(\mathrm{true}\) if \(a+b+c\leq d\) and to \(\mathrm{false}\) otherwise.

Comments

  • 1.

    Its conjoint is the function that sends \((d,a,b,c)\) to \(\mathrm{true}\) if \(d\leq a+b+c\) and to \(\mathrm{false}\) otherwise.

    Comment Source:Its conjoint is the function that sends \\((d,a,b,c)\\) to \\(\mathrm{true}\\) if \\(d\leq a+b+c\\) and to \\(\mathrm{false}\\) otherwise.
Sign In or Register to comment.