elementary set theory - Cartesian product (without order) and operation

I think that I know the formal definition of the Cartesian product of order pairs, but in my example the order doesn’t make sense. Let’s look at an example. I have 4 sets:$A_1=\{ milk, 3\}$, $A_2=\{water, 1\}$, $A_3=\{black, no, 1 \}$, $A_4=\{milk, no \}$The Cartesian product $A = A_1 \times A_2 \times A_3 \times A_4 = \{ $$(milk,water,black,milk)$,$(milk,water,black,no)$,$(milk,water,no,milk)$,$(milk,water,no,no)$,$(milk,water,1,milk)$,$(milk,water,1,no)$,$(milk,1,black,milk)$,$(milk,1,black,no)$,$(milk,1,no,milk)$,$(milk,1,no,no)$,$(milk,...Read more

elementary set theory - Proving that product of mutually disjoint sets is equipotent

The question I am trying to prove is as follows:Let $\{B_i\}_{i \in I}$ and $\{C_i\}_{i \in I}$ be families of mutually disjoint sets. If $B_i \approx C_i$ for each $i \in I$, prove that $\prod_{i \in I}B_i \approx \prod_{i \in I}C_i$.My attempt:Since $B_1 \approx C_1$, and $B_2 \approx C_2$,this implies that $B_1 \times B_2 \approx C_1 \times C_2$and, $B_1 \times B_2 \approx C_1 \times C_2$ and $B_3 \approx C_3$this implies that $B_1 \times B_2 \times B_3 \approx C_1 \times C_2 \times C_3$Repeating this process for all $i \in I$, we get the de...Read more

elementary set theory - Relate a set that is not a cartesian product to cartesian products

Consider 3 sets$$A\equiv \{a_1,a_2,a_3\}$$$$B\equiv \{b_1,b_2,b_3\}$$$$C\equiv \{c_1,c_2,c_3\}$$Let $\mathcal{A}$ denote all possible subsets of $A$ excluding the empty set. Similarly, $\mathcal{B}$ and $\mathcal{C}$. I.e.,$$\mathcal{A}\equiv \Big\{A, \{a_1\}, \{a_2\}, \{a_3\}, \{a_1,a_2\}, \{a_2, a_3\}, \{a_1,a_3\}\Big\}$$$$\mathcal{B}\equiv \Big\{B, \{b_1\},\{b_2\}, \{b_3\},\{b_1,b_2\}, \{b_2, b_3\}, \{b_1,b_3\}\Big\}$$$$\mathcal{C}\equiv \Big\{C, \{c_1\},\{c_2\}, \{c_3\},\{c_1,c_2\}, \{c_2, c_3\}, \{c_1,c_3\}\Big\}$$Some additional definitio...Read more

elementary set theory - Proof of theorem about equivalence classes

I am looking to understand the following theorem, and I am also wondering what is meant by "mutually disjoint", or at least how it's to be understood in the following context: The distinct equivalence classes of an equivalence relation on $A$ provide us with a decomposition of $A$ as a union of mutually disjoint subsets, conversely given a decomposition of $A$ as a union of mutually disjoint, nonempty subsets, we can define an equivalence relation on $A$ for which these subsets are the distinct equivalence classes.I am also looking to know if...Read more

elementary set theory - Intersection of set partitions

I am trying to figure out a good way of finding the intersection of two partitioned subsets of a set (or what to call what I'm trying to do so I can read something about it).Let's say I have two subsets, ABC and BCDE. The first is partitioned into A/BC (part A vs part BC) and the second is partitioned into B/CDE (part B vs part CDE). Whatever the two subsets are, the partitions are always bipartite.Now I want to combine these subsets, retaining the partitions so that I have A/B/C/DE. I may implicitly be treating the fact of the two subsets as a...Read more

elementary set theory - The sets $\{-x,x\}$ form a partition of $\mathbb Z$

I've been really trying to understand how some of these proofs work; I've spent a majority of my time studying the material for this class, but I'm still performing poorly in it. It doesn't help that the book is very vague; what's worse is that it contains little to no solutions and does not have a solutions manual, so I don't even know if I'm right or wrong half the time. Anyways, in the problem, we are asked to prove that a set is a partition. A problem from the book: Prove that $P=\left\{X: X = \{-x,x\} \space \text{and} \space x\in\mathb...Read more

Equivalent definition of partition in set theory

According to my Prof, the definition of partition in Set-theory is$S\subseteq P(A) \smallsetminus\{\emptyset\} $ is partition of A if for All $a\in A$ exists $T\in S$ unique such that $a\in T$.According to How to prove it by Daniel J. Valleman, I see another definition of partition which as following:Suppose A is a set and $F \subseteq P(A) $ F is called a partition of A if it has the following properties:UF = A.F is pairwise disjoint which means for all $X,Y\in F$ if X is not equal to Y then $X\cap Y = \emptyset$For All $X\in F$ X is not $\em...Read more

elementary set theory - Consider P a partition of set A. Given relation R on A and xRy if and only if x, y $\in$ X for some X $\in$ P. Show R is equivalence relation on A

Consider $P$, a partition of a set $A$. Define a relation $R$ on $A$ such that $x\mathrel{R}y$ if and only if $x, y \in X$ for some $X \in P$. Show that $R$ is an equivalence relation on $A$. Next show that $P$ is the set of equivalence classes of $R$. For the first part, proving $R$ is an equivalence relation on $A$: I think I understand how $R$ is reflexive and symmetric. Since $X$ is a set with a partition, every element in that set is related to itself and related to each other so $R$ is by definition reflexive and symmetric. Can I use the...Read more

elementary set theory - Proving $F . G$ is the greatest lower bound

This is one of the problem I have been solving from Velleman's How to Prove book: Suppose $A$ is a set. If $F$ and $G$ are partitions of $A$, then we'll say that $F$ refines $G$ if $\forall X \in F \exists Y \in G(X \subseteq Y)$. Let $P$ be the set of all partitions of $A$, and let $R = \{(F,G) \in P \times P \mid F \text{ refines } G \}$ (...) Skip the first two questions c) Suppose $F$ and $G$ are partitions of $A$. Prove that $F . G$ is the greatest lower bound of the set $\{F,G\}$ in the partial order $R$.Note that $F. G$ is define...Read more

elementary set theory - How to express the set of intersections between two ordered sets by selecting exactly one element per index?

Given a set $\mathcal{S} = \{S_1, S_2, S_3\}$, two ordered sets can be produced:$\mathcal{S}^+$, where $S^+_i \in (S_1,S_2,S_3)$, and$\mathcal{S}^-$, where $S^-_i \in (U\setminus S_1, U\setminus S_2, U\setminus S_3)$, where $U$ is the universal set, i.e., $S^-_i$ is the complement of $S_i$.How can I succinctly define a function $P(S_1,S_2,S_3)$ that produces a set of intersections for all combinations between the sets $\mathcal{S}^+$ and $\mathcal{S}^-$, while selecting exactly 1 element for each index?In the above example, $P(S_1,S_2,S_3)$ wou...Read more

elementary set theory - A parition induced on $A$ by $f^{-1}(Y_1),f^{-1}(Y_2),...,f^{-1}(Y_n)$

The following thought just occured to me on my way back home i forgot it for some while but now i remembered it again is the argument below correct? Proposition. Lets say that we have a surjective map $f:A\to B$ where $A$ and $B$ are aritrary sets. In addition we have the subsets sets $Y_1,Y_2,...,Y_n$ such that they determine a partition over the set $B$. Does it then follows that the sets $f^{-1}(Y_1),f^{-1}(Y_2),...,f^{-1}(Y_n)$ determine a partion over the set $A$.I think that the answer is yes and the following is my attempt at a pro...Read more

elementary set theory - Existence of a denumerble partition.

Is the following Proof Correct?NOTE: In the text prior to the following exercise we have already proved that $\mathbf{Z^+}\times\mathbf{Z^+}$ is equinumerous to $\mathbf{Z^+}$.Theorem. Given that $A$ is denumerable. There exists a partition $P$ of $A$ such that $P$ is denumerable and for all $X\in P$, $X$ is denumerable.Proof. Since $A$ is denumerable there exists a bijection $h:\mathbf{Z^+}\to A$ furthermore from our initial discussion in $7.1$ we know that $\mathbf{Z^+}\times\mathbf{Z^+}\thicksim\mathbf{Z^+}$ consequently there exists a bijec...Read more