Articles of elementary set theory

What does P in blackboard bold type of letter stand for? ℙ?

In the first post of the thread “Cardinal number subtraction”, Cardinal number subtraction there is a symbol for some kind of set which looks like this: ℙ I am familiar with symbols for natural ($\mathbb{N}$), rational ($\mathbb{Q}$), real ($\mathbb{R}$), complex ($\mathbb{C}$) numbers, which are all written in blackboard bold type. I am not a mathematician, […]

Alternate Proof for one-one function

I need an alternate proof for this problem. Show that the function is one-one, provide a proof. $f:x \rightarrow x^3 + x : x \in \mathbb{R}$ I needed to show that the function is a one-one function. I tried doing $f(x) = f(y) \Rightarrow x = y$, It ended up with $x(x^2 + 1) = […]

Minimal Connected Set containing a Closed Connected Set in a Compact Space

This question came from Dugundji’s $\textit{Topology}$: Given a compact, connected space $X$, let $A \subset X$ be closed. Prove that there exists a closed, connected set $B \subset X$ such that $A \subset B$ and any proper subset of $B$ is either not connected, not closed, or does not contain $A$. The text has an […]

Proving the generalized intersection of the interval (0, 1/n) is the empty set?

Prove that the generalized intersection of the interval (0,1/n) is the empty set? Aka prove that $(0,1) \cap (0, 1/2) \cap (0, 1/3) \cap (0, 1/4) … = \emptyset$. I know that I need to prove this by contradiction, by assuming that there exists an x in the intersection and then choosing a positive integer […]

What's the right moment to learn Set Theory?

I’ve seen a question in which the OP asked when is the right moment to learn Category Theory, it seems this moment comes a little after a course of algebra, and indeed some books on abstract algebra brings concepts of Category Theory, such as Jacobson’s Basic Algebra or Paolo Aluffi’s ALGEBRA, Chapter 0. But until […]

Is the collection of finite subsets of $\mathbb{Z}$ countable?

The collection of all subsets of $\mathbb{Z}$ is uncountable, due to Cantor’s theorem But how can I prove that the collection of all finite subsets of $\mathbb{Z}$ is countable?

Inverse of composition of relation

I’m doing preparaton problems for my exam and one of the first problems in the “composition of relations” section is this: Prove: $$ (A \circ B)^{-1} = B^{-1} \circ A^{-1} $$ I know I need to prove 2 inclusions (L = Left side of the equation, R = right side of the equation): $ L […]

Is $\{\varnothing \}$ an empty set?

Is $\{\varnothing \}$ an empty set ? this suppose to be 7th grade math ,i went through the empty set lesson in the textbook , basically i know that {} or $\varnothing$ is an empty set but what about $\{\varnothing\}$ which is a question in the textbook , i was thinking what if $\varnothing$ is […]

Example of an injective function $g$ and function $f$ such that $g\circ f$ is not injective

Give an example of a function $g$ which is injective, but for which its composition with $f$ is not, namely $g\circ f$. I suspect that $f(x)=0$ and $g(x)=x$ will do, am I right?

Proving $a\le f(a)$ for a monotonically increasing function on a well-ordered set

Let $(x,\le)$ be well-ordered set and let $f: \ x \rightarrow x$ be monotonically increasing function. Prove that $\forall a \in x$ $$a \le f(a)$$ Find an example of set $x$ linearly ordered such that the statement doesn’t hold. My try: Assume $X=\{a \in x \ : \ a \ge f(a) \}$ is non-empty set […]