First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic has these variables as well as additional variables that range over sets of individuals. For example, the second-order sentence $\forall P\,\forall x (x \in P \lor x \notin P)$ says that for every set $P$ of individuals and […]

When I am learning, one thing I am puzzled is the definition. For example, we define $0$ as $\emptyset$. But when we use set language, how could we know we are talking about $0$ or the empty set. Another example is the definition of order pair $\{\{a\},\{a,b\}\}$, how do we know we are talking about […]

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, […]

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) = […]

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 […]

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 […]

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 […]

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?

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 ? 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 […]

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