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# Exercise 37 - Chapter 4

edited August 14

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.

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.