Is there a better way to understand set operation
In measure theory, for example, we always need to find a good way to express a set to show its measurability. eg: we write $ A cap B$ as $$ A cap B = Dbackslash( (Dbackslash A) cup (Dbackslash B) ). $$...
View ArticleSchröder-Bernstein theorem and injective maps
I am a bit confused about the statement of the Schröder-Bernstein theorem which states the following: Suppose that $A$ and $B$ are sets, and that $f : A to B$ and $g : B to A$ are injective mappings....
View ArticleStandard notation for 'same' function with different ranges
Does anyone know of a standard notation for the situation when we want to define the ‘same’ function but on a larger or smaller range. More precisely, if $$f:A to B$$ is a function and $C$ contains...
View ArticleIs this directed set countable?
A directed set is a pair $(A,leq)$ where $leq$ is a reflexive, transitive relation such that for any $x,yin A$ we have some $z$ such that $x,yleq z$. (This comes up when dealing with categorical limits...
View ArticleConstruct an injective function $f:[a,b]times[c,d]rightarrowmathbb{R}$ for...
Let $a,b,c,dinmathbb{R}$ such that $a<b$ and $c<d$ be given. Construct an injective function $f:[a,b]times[c,d]rightarrowmathbb{R}$. My intuition is to construct a function...
View Articlereal analysis proof related to functions
It is asked to prove that, let $f$ be a one to one function if and only if A,B are sub sets of real number $f(A⋂B) = f(A)⋂f(B)$. I have if $f$ is one to one then the results. but I find it hard to...
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Four mutually exhaustive sets; finding the intersection of two sets. Set...
In a group of 120 persons there are 80 elements of B and 40 elements of G. Further 70 persons in the group are M and the remaining are H. Then the number of elements that are both in B as well as M. I...
View ArticleOrdinal Fractions
Is any fraction $,{x}big/ {y},$ an ordinal number and if so, does ordinal $,1 = biglbrace0,dots,y – 1big/ybigrbrace,$ instead of $,leftlbrace0rightrbrace,$? “If (X, <=) is a well ordered set with...
View Article90% of students face problems in A, 80%in B, 75%in C and 70% in D. What is...
The complete question is In a school , 90% of students face problems in A, 80%in B, 75%in C and 70% in D. What is the minimum percentage of people facing problems in all the subjects. It looks fairly...
View ArticleInfinite sets and equipotence
I need to prove that: if A is an infinite set and x is some element such that x is not an element of A, then (A union {x}) is equipotent to A. The thing is I know it’s relatively easy to prove it with...
View ArticleExistence of certain set triple in certain set system
Let $mathcal{F}$ be a family of subsets of some finite set $X$, where $left| Xright|geq 5$, with the property that for every $a, b, cin X$ there exists $Finmathcal{F}$ with $Fsubseteq Xbackslash{a, b,...
View ArticleAre computable sets closed under XOR?
How do I prove if computable are/are’t closed under XOR?
View ArticleProving if a relation is or is not transitive.
Let S be a set that contains at least two different elements. Let R be the relation on $P(S)$, the set of all subsets of S, defined by $(X, Y ) in R$ if and only if $X cap Y = emptyset$. I am trying to...
View ArticleWhat is the intersection of the sets ${1}$ and ${1,2}$?
What is the intersection of the sets ${1}$ and ${1,2}$? For me, it would make sense that ${1} cap {1,2} = 1$, but I’m afraid it must be ${1}$, otherwise for instance $T = { {}, {1}, {1,2} }$ would not...
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...
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 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 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 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 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,...
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