Infinite 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 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 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...
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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 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...
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 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 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 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 ArticleIs 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 $$ A cap B = Dbackslash( (Dbackslash A) cup (Dbackslash B) ). $$...
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