Equivalence classes and relations
Consider the equivalence relation defined on the unit disk $D^2={(x,y) in mathbb R^2: x^2 + y^2 le 1}$ by $$(x_1,y_1)~ (x_2,y_2)Leftrightarrowtext{ either }(x_1,y1)=(x_2,y_2)text{ or }x_1^2...
View ArticleThe minimum cardinal of a geometrical set
Let $S$ be a set of points in a plane $P$, having the following property: for any point $X in P$ there is at least one point $M in S$ so that the distance $|XM|$ is rational. Find the minimum cardinal...
View ArticleHow do I disprove a set is not a subset of another set?
Given that $T={3t|t inmathbb Z}$, $Q={5q|q inmathbb Z}$, $R={6r|r inmathbb Z}$ and $S={T,Q,R}$. How can I disprove that Q $subseteq$ R? I tried the following: Let $q=2, Q={10}$ Let $r=2, R={12}$,...
View ArticleTricky Probability! (or impossible)
Suppose that we have 3 independent events (Students that they are attending lectures) P(a) = .10 P(b) = .17 and P(c) = .23 What is the probability that during a school day that two of these events...
View ArticleDoes the usual law for image also hold for relations?
Let $R subseteq U times V$ be a relation, and $S_0, dots, S_{m-1} subseteq U$ Then does the following hold? $$Rleft(bigcap_{j=0}^{m-1} S_j right) subseteq bigcap_{j=0}^{m-1} R(S_j)$$ It is easily...
View ArticleGiven that something holds for an arbitrary union of sets, does it also hold...
In a problem I am working on, I want to say the following: Since $int f chi_{[0, 1] setminus cup_{i = l}^{infty}I_{n_i}} = 0$ for each $l$, $int f chi_{[0, 1] setminus cap_{l = 1}^{infty}cup_{i =...
View ArticleWhat is the difference between an indexed family and a sequence?
For indexed family wikipedia states: Formally, an indexed family is the same thing as a mathematical function; a function with domain J and codomain X is equivalent to a family of elements of X indexed...
View ArticleIdentifying a sequence as subset of subspace
If I have some sequence $mathcal A = (a_i)$ of objects $a_i$ (maybe finite, maybe countably infinite) how can I say that those objects all exist in some subspace $S$? Is it correct to say $mathcal A...
View ArticleCartesian Product Proof for Three sets
I have a homework question I need help with. I need to show that if $A times B$ is a subset of $B times C$ then prove A is a subset of C.
View ArticleQuick Clarification: Definition of Bijective Function
I am very familiar with the concepts of bijective, surjective and injective maps but I am interested in improvising the definition of bijection in a way I have not seen done before. To be clear I will...
View ArticleClarify definitions of relation and 0-ary relation
From mathworld.wolfram.com: A relation is any subset of a Cartesian product But if so, then the null set is all of: 0-ary, 1-ary, 2-ary etc. Wouldn’t it be better to define it as: A relation is any...
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Inverse image of $[0,1/2]$ for $f(x)=sin(x)$
Let $f:mathbb{R}rightarrowmathbb{R}$ be defined by $f(x)=sin(x)$. Describe the set: $$f^{-1}bigg(Big[0,frac{1}{2}Big]bigg).$$ My answer is $$f^{-1}([0,1])=bigcup_{ninmathbb{Z}}...
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Problems with proving two sets are equal
Can someone help me with this? I tried solving it but I got stuck
View ArticleIs $G^{(X, Y)} = (G^X)^Y?$ ($A^B$ just means that $B$ is mapped to $A$)
So I have a set $G={banana,potato}$.To do $G^{(X,Y)}$ ($A^B$ just means that map $(x,y)$ to $G$),I map a Cartesian product $(x,y)$ to every element of $G$,which produce...
View ArticleI can't understand the formal definition of $mathbb{R}$
I’ve always intuitively understood this set in intuitive sense, as “all numbers on the number line”. However, now I want to know the formal definition: Consider the set of rational numbers,...
View ArticleWhat does it mean for the empty set to be connected and totally disconnected?
I am trying to prove that the empty set is disconnected, but every single post I can find on this topic is about showing empty set is connected. Recall definition of connected. A set $S$ is connected...
View ArticleUppercase E notation for sets?
In Jónsson and Tarski’s (1951) paper Boolean Algebras with Operators, Part I from the American Journal of Mathematics, they write formulae such as $L_i = underset{u}{mathbf{E}} , [u in At^m text{ and }...
View ArticleIs my proof of the principle of backward induction using well-ordering correct?
I’m trying to prove backward induction, which I’ll state as follows: Consider the set $mathsf{A}$, where $nin{mathsf{A}}$, and $m+1in{mathsf{A}}$ $implies$$min{mathsf{A}}$. Then $mathsf{A}={0,…,n}$....
View Articlesets of natural numbers [duplicate]
Possible Duplicate: Countable set having uncountably many infinite subsets Question: Is it possible to find uncountably many infinite sets of natural numbers that any two of these sets have only...
View ArticleLargest possible value of $P(A cap B)$
Suppose $A$ and $B$ are events with $P(A)+P(B)>1$. Show that the largest possible value of $P(A cap B)$ is $ min(P(A), P(B))$. I suspect I’m supposed to use $P(A cap B) = P(A)+P(B) -P(A cup B)$ but...
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