# Questions tagged [elementary-set-theory]

For elementary questions on set theory. Topics include intersections and unions, differences and complements, De Morgan's laws, Venn diagrams, relations and countability.

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### Would this be a valid set definition for $Graph(f)$?

I had this particular set in mind: $G_f=\{(x+y, y):x \in \mathbb{R}$ and $y=f(x+y)\}$ or $G_f=\{(x+y, y):x+y \in \mathbb{R}$ and $y=f(x+y)\}$ or $G_f=\{(x+y, y):x,y \in \mathbb{R}$ and $y=f(x+y)\}$ ...
1 vote
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### $\sigma$-algebra on $\mathbb{R}$ generated by the collection of all one-point sets.

I have seen that the $\sigma$-algebra on $\mathbb{R}$ which is generated by the collection of all one-point sets is the collection of all sets with a countable number of elements and their complements,...
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### Why is the set $A$ not closed under countable unions?

Let $A$ be the set of all finite unions of half open intervals of the form $(a,b], a,b \in \mathbb{R}$, $(-\infty, b]$ or $(a,\infty), a \in \mathbb{R} \cup \{-\infty\}$. I am reading that this set is ...
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### Notation for map evaluated at all elements of a set?

Consider a map $f : X \to Y$ and some given set $A \subseteq X$. I would like to introduce the notation $f(A) = \{ f(a) \vert a \in A \} \subseteq Y$, i.e. the set of output given the elements of $A$ ...
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### Set theoretic definition of terms of the untyped lambda calculus

I am trying to translate the following definition (in Agda) of intrinsically scoped terms of the untyped lambda calculus into more mathematical (in particular set theoretical) notation: ...
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### Proof of the principle of induction [duplicate]

I will be referencing the proof I provided here. I don't understand the remark of a person, who states the incorrectness of such proof. From what I could understand, the proof is also valid not ...
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1 vote
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### Simplification of $(X_1 \times X_2) \setminus ((X_1 \setminus U) \times (X_2 \setminus V))$, $X_1, X_2$ infinite and $U \subset X_1, V \subset X_2$

I am trying to derive a simplification of this set equation. I am trying to work through it "intuitively," but I am stuck and am wondering of any set theory that can help me simplify this ...
1 vote
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### Interpreting statements in a proof which have implications for the non-emptiness of sets (sets, logic, cartesian products) - Tao Analysis I 3.5.6

This question is about deriving requirements for non-emptiness of sets from the algebraic steps in a proof. It is not primarily about the proving the primary objective of the exercise. Why am I asking ...
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### Demonstrate that A is a countable set.

Question: Let $A = \{x \in \Bbb N \mid \exists y(x = 2y \lor x = y^2)\}$. Construct a surjective mapping $f: \mathbb{N} \rightarrow A$. By doing this, you demonstrate that A is a countable set. ...
1 vote
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### Expressing the event $\{ \limsup_{n \rightarrow \infty} X_n = l \}$

Let $(\Omega, \mathcal{H}, \mathbb{P})$ be a probability space. Let $(X_n)_{n=1}^{\infty}$ be a sequence of real valued random variables s.t. $\forall n \geq 1 \quad X_n$ is $\mathcal{H}$-measurable. ...
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### Show that the interval ⟨2, 5⟩ ⊆ ℝ⁺ is an uncountable set.

Question: Show that the interval $⟨2, 5⟩ ⊆ ℝ⁺$ is an uncountable set.\ To show that the interval $\langle 2, 5 \rangle \subseteq \mathbb{R}^+$ is an uncountable set, we can use Cantor's diagonal ...
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### Finite cardinals raised to the power of an infinite cardinal

I am trying to prove the fact that if $a$ and $b$ are finite cardinals, and $c$ is an infinite cardinal, then $a^c = b^c$. I am able to prove this fact by using $d \cdot d = d$ for all infinite ...
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### Szekeres proof of set distributive law inadequate?

I'm reading "A Course in Modern Mathematical Physics" by Peter Szekeres. In problem 1.1, he asks the reader to show the distributive law $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$. The ...
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### proof of part of Tao Analysis I 3.5.6 too simple? (sets, cartesian products, logic)

I came up with a simpler solution to one part of Tao's Analysis I 4th ed exercise 3.5.6. Question: Several online solutions, and the one I originally developed (with help), use proof by contraposition,...
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### How to describe the transitive closure of a relation in terms of the ground relation

Let $X$ be a non-empty set and $\equiv$ a relation on $X$, which is symmetric and reflexive but is not transitive. I know that there exists a transitive closure of $\equiv$, saying, $\equiv_{cl}$. But ...
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### Can we refer to "infinity" as a property of a set in ZFC/FOL?

I understand we have the Axiom of Infinity and have "access" to an infinite set, but is there a way to say in first-order logic "set A is infinite" or refer to its cardinality? The ...
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### Axiomatic set theory: problems understanding union of a set $\bigcup X$

From Jech p 9 (Set Theory, 3rd edition ) if $X$ is a set then $Y=\bigcup X$ is a set which is stated in formulas as; \begin{align*} \forall X\exists Y\forall u\in Y\leftrightarrow\exists z(z\in X\land ...
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