# 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.

20,011 questions
Filter by
Sorted by
Tagged with
36 views

### Prove that for any set A and B, the cardinality of the set of all functions mapping A to B is $\vert B \vert ^ {\vert A \vert}$

What I do is for finite sets A, B, let $A={a_1, a_2, ...a_n}$ and $B={b_1, b_2, ...b_m}$ A function f assigns each element $a_i$ of $A$ to an element $b_j = f (a_i)$ of $B$; there are $m$ ...
11k views

### How to prove that $\mathbb{Q}$ ( the rationals) is a countable set

I want to prove that $\mathbb{Q}$ is countable. So basically, I could find a bijection from $\mathbb{Q}$ to $\mathbb{N}$. But I have also recently proved that $\mathbb{Z}$ is countable, so is it ...
46 views

63 views

### How to prove that a function $f:X\to X$ is injective if and only if surjective?

I intuitively understand it and can prove for finite sets. But can you prove it? And my proof relies on counting the elements. Can you show it without counting or using cardinalities?
66 views

### What is the cardinality of $\Bbb Q ^4$?

What is the cardinality of $\Bbb Q ^4$? I understand that $\vert \Bbb Q \vert=\aleph _0$, and $\vert \Bbb Q ^4 \vert$ supposed to be $\aleph _0$, is there any formal proof to this?
23 views

### Sets re: equality and subsets

The italics are my comments. This is a proof from a text on abstract algebra that seems a bit ambiguous. But I'm just learning sans instructor. My question is: is there a convention in set theory ...
28 views

### The images of two arbitrary functions can partition their domains

The source of this problem is from Dugundji's Topology. Let $f\colon X \to Y$ and $g\colon Y \to X$ be any two maps. Show that $X$ and $Y$ can each be expressed as disjoint unions: $X = X_1 \cup X_2$ ...
34k views

### Produce an explicit bijection between rationals and naturals?

I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but ...
26 views

83 views

### What is the cardinality of $A$ if $A = \{\{A\}\}$? [on hold]

What is the cardinality of $A$ if $A = \{\{A\}\}$? I think one is the answer since the outer set has one element, a set. But that seems to simple to be right.
32 views

### Property of Set Function related to Supermodularity

Have set functions with the following property been studied? If so, what is the name of the property? Let $\Omega$ be a set of elements, and let $f:\mathcal P(\Omega) \rightarrow \mathbb{R}$ be a ...
27 views

### How to represent a set of variables are different elements of a set mathematically?

Lets say that we have set $A$. The variables $m_{1}$, $m_{2}$, $\cdots$ $m_{N}$ are elements of the set $A$ but with exclusivity. The variables have to be different elements of the set. How can I ...
37 views

### Defining sets in intuitionistic logic

I'm somewhat familiar with the school of intuitionistic logic. I know that an intuitionistic logician thinks of infinity as constructive as apposed to complete. Thus a intuitionistic logician cannot ...
50 views

### Proof verifications for set equality

Can I get a proof verification? Are there any flaws in this proof? The examples in the book are only for sets bounded either above or below. Prove $$\cup_{n\in \mathbb{N}}(0,\frac{n}{n+1})=(0,1)$$ ...
408 views

### Find minimum number of customers that must have visited the bakery that day?

A bakery sells three kinds of pastries -pineapple, choclate and black forest. On a particular day, the bakery owner sold the following number of pastries : $90$ pineapple , $120$ chocolate and ...
206 views

### How do I define this subset using mathematical notation?

Assume $P = \{2, 3, 5, 7, 11, 13, 17, 19, 23,....\}$ or in another words, P is the set consisting of all prime numbers. Now, suppose we want to form the set $S$, which is subset of $P$,and whose ...
42 views

27 views

### Defining subgraph and subset of directed graph

Following the lines of this question Need help with Graph notation for a subgraph , I am trying to define the subset of vertices connected to a specific vertex $v_i$ and need some help working through ...
45 views

### Proving two sets of even integers are equal

This is a question I found in the proof book I am practicing in. This seems like a ridiculous thing to prove, so I don't know what the author is looking for. Prove $\{n \in \mathbb{Z}:n$ is even$\}$...
63 views

69 views

### Prove that a set cannot have two different size $𝑚$ and $𝑛, 𝑚≠n$.

We define the number of elements in set by bijections as follows: $|X| = n$ means that there exists a bijection from X to the set $\{1,2 \dots, n\}$. I already showed that: if $X$ and $Y$ have ...
83 views

### A Set is Infinite if, and only if, it is in One-to-one Correspondence with a Proper Subset of Itself

Can someone explain what that means? How can there exist an injective function from an infinite set to a proper subset of itself. A function from a set A to a set B where B has fewer elements than A ...
23 views

### Describe the set $\{ z\in \mathbb{C}: |z^2 -1|<1\}$ [duplicate]

Let $z=a+ib$ $|z^2-1|<1$ $(z^2-1)(\overline{{z^2-1}})<1$ $z^2\overline{z^2}-z^2-\overline{z^2}<0$ $|z^2|^2<2Rez^2=a^2-b^2$ What can we say further? Is there any mistake? Is it ...
34 views

### Unique Equivalence Classes

Not sure if this is a suitable question here, but I'm having trouble understanding the intuition behind a theorem I've read in a textbook. So it says the following: "If $\mathscr R$ is an ...
137 views

### What is the justification for calling a hereditary system an independent system?

I was learning about set systems and hereditary systems and I noticed that they also call a hereditary system a independence system and that didn't quite make sense to me intuitively. First recall ...
55 views

### Confusion about Notation of the Cardinality of a Set

One textbook I'm reading says that the definition of two sets having the same cardinality is as follows: "Two sets A and B have the same cardinality if there exists a bijection $f:A \rightarrow B$." ...
63 views

### What is the Simplest Explanation for the Countability of the Integers?

What is the simplest (or at least simple to understand) if one wanted to explain why the set of Integers has the same cardinality as the set of natural numbers to students who have a vague idea of why ...
32 views

### What does the symbol “\” mean in the context of set operation?

I am learning group by using this post let's call this figure_1. everything is OK until the set of B shows up. what I already known is The union is notated A ⋃ ...
I'm trying to create a sequence of elements which are dependent on their position in the sequence. What I mean is that I want the value $r$ in position 1 equal to 1, on position 2 equal to 2 and so on....