These were pulled from Exercises for $\S5$ of Chapter I in Bourbaki’s set theory book. (Assume $\mathfrak{T}$ has no constants, i.e. no letters in explicit axioms; thus we can abuse I.2.3.C3 without repercussions) Exercise 1.Solution.Exercise 1. Show the relation $x=y$ is functional in $x$ in $\mathfrak{T}$. Solution. Denote the relation $x=y$ by $R$. Let $a$ and…
These were pulled from Exercises for $\S$3 and Exercises for $\S$4 of Chapter I in Bourbaki's set theory book. Exercise 1.Solution.Exercise 1. Let $A$ be a relation in logical theory $\mathfrak{T}$. If $A\iff (\mbox{not } A)$ is a theorem in $\mathfrak{T}$, show that $\mathfrak{T}$ is contradictory. Solution. The negation of $A\iff (\mbox{not } A)$ is…
These were pulled from Exercises for $\S$2 of Chapter I in Bourbaki's set theory book. Exercise 1.Solution for a.Solution for b.Exercise 1. Let $\mathfrak{T}$ be a theory, let $A_1,A_2,...,A_n$ be its explicit axioms and $a_1,a_2,...,a_h$ its constants. Let $\mathfrak{T}'$ be the theory with signs and schemes are the same as those of $\mathfrak{T}$, and whose…
These were pulled from Exercises for $\S$1 of Chapter I of Bourbaki's set theory book. Exercise 1.Solution.Exercise 1. Let $\mathfrak{T}$ be a theory with no specific signs. Show that no assembly in $\mathfrak{T}$ is a relation and that the only assemblies in $\mathfrak{T}$ which are terms are assemblies consisting of single letters. Solution. Let $A$…