# Questions tagged [elementary-set-theory]

This tag is for elementary questions on set theory, spanning topics usually found in introductory courses in set theory, in addition to review sections of graduate textbooks in the same field. Topics include intersections and unions, differences and complements, De Morgan's laws, Venn diagrams, relations, etc. More advanced topics should use (set-theory) instead.

19,257 questions
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### Prove a function is onto if its domain is a Cartesian product

I've been working on this problem: Suppose the function $f:\mathbb Z \times \mathbb Z \to \mathbb Z$ is defined by $f(n,m)=2nm-1$. Is this function onto? After a while, I figured out that $4$ can'...
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### Cardinality of the unit square and union of sets of size $c$ are equal

So this is a simplified version of the theorem where union of sets of cardinality $c$ has cardinality $c$. $c$ refers to the continuum. We try to prove instead the union of sets of cardinality $c$ ...
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### $\omega +1$ is not isomorphic to $\omega$ (in the well-ordering by $\varepsilon$)

$\omega +1$ is not isomorphic to $\omega$ (in the well-ordering by $\varepsilon$). I see that $\omega +1$ does have maximal element but $\omega$ is not so there is no ismorphism between $\omega +1$ ...
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### Why does $A_j \subseteq cl ( \bigcup_{m=1}^n A_m )$ imply $cl(A_j) \subseteq cl ( \bigcup_{m=1}^n A_m )$?

Why does $A_j \subseteq cl \bigg( \cup_{m=1}^n A_m \bigg)$ imply $cl(A_j) \subseteq cl \bigg( \cup_{m=1}^n A_m \bigg)$? This is intuitive, but I was thinking of whether one can be sure that there are ...
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### Help with the proof of Theorem 1 (Chapter 2) of Suppes' “Axiomatic Set Theory” [on hold]

See the proof of Theorem 1 : $x \notin 0$ page 21 and page 22. Anybody can help me with the first step of the proof. I don't understand why the author uses in this step "x belongs to empty set"...
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### Help proving the following using Archimedean Property of Reals [duplicate]

For each $n=1,\,2,\,3,\dots$, let $D_n = \displaystyle{\left(-\frac{1}{n},1+\frac{1}{n}\right)}$. Prove that $\displaystyle{\bigcap_{n=1}^\infty D_n} = [0,1]$. Hello, I've been stuck on this problem ...
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### Syllogism question using set theory notation

i'm stuck on a question where I have to explain the set theory notation of the syllogism below; boats are vessles $(A ⊆ B)$ boats operate in water $(A ∈ C)$ or $(A ⊆ C)$ Some vessles operate in ...
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### If $S$ is countable then $S_0 \subset S$ is countable.

Define "countable" in the following way: $S$ is said to be countable if $S$ is finite OR $|\mathbb{N}| = |S|$. So my textbook proves the theorem by considering two cases. Case 1: $S$ is finite. ...
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### Is the set of all finite sets of powers of 2 countable?

Sort of confused on how to approach this question. I know that the set of powers of 2 is infinitely countable and the set of all sets of powers of 2 is the power set which isn't countable because it's ...
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### Is the multivariate function $G:\mathbb{Q}$ x $\mathbb{Q} \rightarrow \mathbb{R}$ where $G(x, y) = r + \sqrt 2 * s$ onto, one-to-one, or both?

I recently had a quiz in which I was given the following question: Define a function $G:\mathbb{Q}$ x $\mathbb{Q} \rightarrow \mathbb{R}$ where $G(x, y) = r + \sqrt 2 s$. Is G onto, one-to-one, ...
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### Is $\bigcup\limits_{i=1}^{\infty} \left(-\frac{1}{i},\frac{1}{i}\right)$ finite?

In a practice exam, there is a question asking if $$\bigcup\limits_{i=1}^{\infty} \left(-\frac{1}{i},\frac{1}{i}\right)$$ is finite, countably infinite, or uncountable. The solution to the practice ...
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### Can we expand “induction principle” to a partial order $(X, \leq)$?

We know that every infinite can be made well-ordered with an unknown order. Also we can expand the induction principle on any infinite set in the sense that it can made well ordered. Now partially ...
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### Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$? [duplicate]

Is there any bijection between $\mathbb{R}$ and $\mathbb{R}^2$ ? If have then what is the mapping ? Please define the mapping. They have same cardinality then it is possible to have a bijection ...
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### Let $A$ be a finite set. Prove that there exists an $f:n \to A$ that is onto $A$ for some $n \in \omega$

Let $A$ be a finite set. Prove that there exists an $f:n \to A$ that is onto $A$ for some $n \in \omega$. Here by finite we mean: A set $X$ is finite iff there is a one-to-one function $f:X→n$ for ...
### Let $n \in \omega$. Suppose $f:n \to A$ is onto $A$. Prove that $A$ is finite.
Let $n \in \omega$. Suppose $f:n \to A$ is onto $A$. Prove that $A$ is finite. I have: Let $I_a = \{i \in n:f(i)=a\}$ for $a \in A$. Since $f$ is onto $A$, $I_a$ is nonempty, and by the well-...