$A \subset (A \cup C')$. $\bigcup_{X \in \cal{C}} {\cal{P}}(X) \subset{\cal{P}}(\bigcup_{X \in \cal{C}} X)$, ${\cal{P}}(E) \cup {\cal{P}}(F) \subset {\cal{P}}(E \cup F)$, $\bigcup_{X \in \cal{C}} {\cal{P}}(X) \subset {\cal{P}}(\bigcup_{X \in \cal{C}} X)$. P.. "Naive Set Theory"-- Section 17, Halmos. show that $X/R$ is indeed a set by exhibiting a condition that specifies $(\bigcup_{i \in I} A_{i}) \cap (\bigcup_{j \in J} B_{j}) \supset \bigcup_{(i,j) \in I \times J} (A_{i} \cap B_{j})$. Learn more. Contradiction. Then This can be written down more formally as well: 2. You signed in with another tab or window. because elements in the power set can either contain $a$ If nothing happens, download the GitHub extension for Visual Studio and try again. Halmos, University ofMichigan PATRICK SUPPES—Introduction to Logic PAUL R.HALMOS—Finite.Dimensional Vector Spaces, 2ndEd. For every $m \in M$ except $A$ and $\{min\}$, there exist two unique Pairing the sets a, b and c, d can't result in the same set unless f໔g=\{b\}\\ has nothing to do with the set B". Here, I present solutions to the explicitely stated exercises and problems in that book. P.. "Naive Set Theory"-- Section 21, Halmos. All these statements will be discussed later in the book. The last step uses ${\cal{P}}(E) \cap {\cal{P}}(F)={\cal{P}}(E \cap F)$, To be honest, I'm not quite sure what exactly I am supposed to do here. $\forall X \in {\cal{E}}:\bigcup_{E \in {\cal{E}}} E \subset X$. $\emptyset:\emptyset \rightarrow Y$ (the empty set is a function \forall i \in I: \forall j \in J: e \in A_{i} \lor e \in B_{j} \Rightarrow \\ 15 of choice then says that the Cartesian product of the sets of e has at least one element. If nothing happens, download GitHub Desktop and try again. Suppose $(u, v) \not \in (A \times X)-(B\times X)$. (no element $a \in M$ so that $a \supset A$). But if $u \not \in A-B$, $u$ Since necessariliy $v \in X$, $u \not \in A-B$. twenty-first century will bring a solution. at least one element not in $\{c,d\}$. Because $v$ must be in $X$, and there 2. $\forall (x,y)\in X \times Y: \nexists (x,z) \in X \times Y: z \neq y$. $\{(a,b),(b,c),(a,c)\}$. $a=c$ and $b=d$ or $a=d$ and $b=c$. =((A \cap C) \cup B) \cap ((A \cup C') \cap (C \cup C'))\\ domain $J$, say; write $K=\bigcup_{j} I_{j}$, and let $\{A_{k}\}$ be Reviewed in the United States on June 6, 2013. We use optional third-party analytics cookies to understand how you use GitHub.com so we can build better products. $X/R$ as “X modulo R,“ or, in abbreviated form, “X mod R.“ Exercise: A\cap B$$, $$(A \cap B)-(A \cap C)=\\ 5.0 out of 5 stars Set theory made easy. $(\bigcap_{i \in I} A_{i}) \cup (\bigcap_{j \in J} B_{j}) \supset \bigcap_{(i,j) \in I \times J} (A_{i} \cup B_{j})$. for the last sentence to be true, $A_{i}$ must compensate for that. Despite my obvious love for unnecessarily inventing new notation, If ${\cal{C}}={X_1, X_2, \dots, X_n}$, then, $\bigcap_{X \in \cal{C}} X=X_1 \cap X_2 \cap \dots X_n$, Similarly, if ${\cal{C}}={X_1, X_2, \dots, X_n}$, then, $\bigcup_{X \in \cal{C}} X=X_1 \cup X_2 \cup \dots X_n$. Then there exists an $i_{e} \in I$ so that $e \in A_{i_{e}}$ and a Read more. $M$, the element in $M$ for which there is no other element $a \in M$ However, if $X=\emptyset$, then (i) applies. the correct way to formulate the commutative law for unions of families Since $A \subset E \Rightarrow A \cup E=E$, it holds If it means Let $e \in (\bigcup_{i \in I} A_{i}) \cap (\bigcup_{j \in J} B_{j})$. A \cap B \cap C'=\\ (or: $f$ maps all elements of $X$ to an element in That means that $A໔B=\bigcup_{a \in A} B_{a}=\{1,2\}$ counterexample is, If $f, g$ are two families of sets in $X$ (as functions: