How are functions written as ordered triples?
In a function, I know that domain $A$ and codomain $B$ are not restricted to sets. They can be proper classes. In that case, how can we write functions as ordered triple? I used to think that in order...
View ArticleShow that any bounded, non-empty set of natural numbers has a maximal...
Help me out this problem. Kindly site some examples because i am a beginner for calculus.
View ArticleSet Theory: Transitive?
I have a question regarding Relations on Sets. Here is the problem: Let $S=left{ a, b, cright}$. Then $R=left{ (a,a), (a,b), (a,c)right}$. Which of the properties (reflexive, symmetric, or transitive)...
View ArticleUsing the well ordering principle to prove a certain property of an integer
The Well ordering principle states that A least element exists in every non empty set of positive integers Use the well Ordering principle to prove the following statement ‘ Any nonempty subset of...
View Article$A cap B subset (A cap C) cup (B cap C')$
How do i show this? $A cap B subset (A cap C) cup (B cap C’)$ $$ x in A cap B$$ $$ implies x in A cap B cup emptyset$$ $$ implies x in A cap B cup (C cap C’)$$ $$ implies x in (A cap B cup C)cap(A cap...
View ArticleDetermine whether, for any set A, it is true that $P(bar{A})= P(U) – P(A)$
I’m currently working on a easy enough logic question, however I’m having trouble proving or disproving it’s validity. The question goes as follows: Determine whether, for any set A, it is true that...
View ArticleConsider the 1000-element subsets
Consider all 1000-element subsets of the set $A = { 1, 2, 3, … , 2015 }$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $frac{p}{q}$, where $p$...
View ArticleProof of $f^{-1}(B_{1}setminus B_{2}) = f^{-1}(B_{1})setminus f^{-1}(B_{2})$
I want to prove the following equation: $$ f^{-1}(B_{1}setminus B_{2}) = f^{-1}(B_{1})setminus f^{-1}(B_{2}) $$ Is this a valid proof? I am not sure, because at one point I am looking at $f(x) in B_1$,...
View ArticleThere exists a bijection between $(0,1)$, $(0,1]$ and $[0,1]$? [duplicate]
This question already has an answer here: How to define a bijection between $(0,1)$ and $(0,1]$? 5 answers
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Can someone present a visualization of the partitioning of a $L^p$ space into...
I am a bit confused by what it means for a $L^p$ space to be partitioned into equivalent classes instead of functions. I understand that give two or more functions $f$, $g$, $h,ldots$ of which are...
View ArticleIntersection of Countably Infinite Sequence of Sets [closed]
Suppose ${Omega_k}_{k=1}^{infty}$ is a sequence of sets, where $Omega_k$ is countably infinite and $Omega_{k+1}subsetOmega_k$ for all $k$. Is it possible to show that $cap _{k=1}^{infty} Omega_k$ is...
View ArticleWhy is AC needed for $|bigcup X_i|=|bigcup Y_i|$, $forall i$ $|X_i|=|Y_i|$,...
On Page 60, Set Theory, Jech(2006), 5.9 If ${X_i : i in I}$ and ${Y_i : i in I}$ are two disjoint families such that $|X_i| = |Y_i|$ for each $i in I$, then $|cup_{i in I}X_i| = |cup_{i in I}Y_i|$ [Use...
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Prove that if $mathcal{F} subseteq mathcal{G}$ then $cupmathcal{F} subseteq...
Suppose $mathcal{F}$ and $mathcal{G}$ are families of sets. Prove that if $mathcal{F} subseteq mathcal{G}$ then $cupmathcal{F} subseteq cupmathcal{G}$ My attempt: Given $mathcal{F} subseteq...
View ArticleCardinality of the Cartesian Product of Two Equinumerous Infinite Sets
Is the cardinality of the Cartesian product of two equinumerous infinite sets the same as the cardinality of any one of the sets? I couldn’t find this explicitly stated in any handout or text. This...
View ArticleAs of August 2015, is the “set” of all gold medalists in the 2016 Olympics a...
As of August 2015, is the “set” of all gold medalists in the 2016 Olympics a set? I think it is since the defining property is very clear. However, given any $x$, we do not know if $x$ is in this “set”...
View ArticleShow that f is surjective
So im having a little trouble proving this. Can anyone help me out? Let $A$, $B subseteq E$. Moreover, let $$f: mathscr{P}(E) to mathscr{P}(A) times mathscr{P}(B)$$ be defined by $$f: X mapsto (A cap...
View Article“Either A and B is open, then A + B is open” (typo sense-making, Stein...
Please advise about the most reasonable way to read this statement. My interpretations are below. The authors do not define the set operation A + B; I assume A + B = $A cup B$. Their statement “Show...
View ArticleExplicit Description for an Equivalence Relation
Given a set function $f : X to X$ let $sim$ be the equivalence relation $x sim f(x)$. Contextually, I am working with the coequalizer of $f$ and $1_X$. I want to have as much information about the set...
View ArticleWhat phenomenon is this? $(2Bbb{Z} + 1)cup 3Bbb{Z} = 2Bbb{Z} cup 3Bbb{Z} + 3$
$(2Bbb{Z} + 1)cup 3Bbb{Z} = 2Bbb{Z} cup 3Bbb{Z} + 3$ Proof: $$ begin{align*} 2Bbb{Z} &= bullet circ bullet circ bullet circ bullet circ dots \ 3Bbb{Z} &= bullet circ circ bullet circ circ...
View ArticleDo De Morgan's laws hold for arbitrary infinite expressions with union and...
It can be shown that De Morgan’s laws hold for infinite union and infinite intersection: $$ left( bigcup_{i in I} A_i right)^c = bigcap_{i in I} A_i^c tag{1} $$ $$ left( bigcap_{i in I} A_i right)^c =...
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