This tag is for elementary questions on set theory - the sort of material covered in "Chapter 0" of graduate texts and in undergraduate set theory texts. Topics include intersections and unions, de Morgan's laws, Venn diagrams, relations, functions, countability and uncountability, power sets, ...

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1
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4answers
35 views

Set builder of this set 0, 1, 3, 6, 10, 15

I have tried to create the set builder of this infinite set: 0, 1, 3, 6, 10, 15, 21, 28,... I have notice that n = (n - 1) + (N + 1) where ...
2
votes
1answer
31 views

Could you give an example of an injective function $f:\mathbb{Z_+}^n\rightarrow \mathbb{Z_+}$ for an integer $n$ s.t. $2\leq n$?

We know that both of the domain the the co-domain are countable sets, so there is a bijection between them, Is there any SIMPLE injection? Here is some injection which I thougt of, but It turns out ...
0
votes
2answers
20 views

Does inclusion-exclusion formula holds for coutanble index set?

Does inclusion-exclusion formula holds for countable index set? Here is the formula for index set of size 2. \begin{align} P(A \cup B)=P(A)+P(B)-P(A\cap B) \end{align}
0
votes
1answer
22 views

Alternative ways to prove an easy set relation

I have a simple set relation, which is almost trivial to prove, but surprisingly, I can only prove it with an "indirect" method, which is bugging me: Let the set $L$ be a subset of $\mathbb{R}^n$. ...
0
votes
0answers
15 views

If every infinite subset of $S$ has an accumulation point in $S$, then $S$ is bounded

If every infinite subset of $S$ has an accumulation point in $S$, then $S$ is bounded. Proof: Suppose $S$ is unbounded. then , for every $m >0,~~\exists~~x_m \in S$ s.t. $|x_m|>m.$ The ...
-1
votes
0answers
28 views

Is $\bigcup_i A_i \backslash \left(\bigcup_i B_i \right)= \bigcup_i (A_i \backslash B_i)$? [on hold]

Is $\bigcup_i A_i \backslash \left(\bigcup_i B_i \right)= \bigcup_i (A_i \backslash B_i)$ if $B_i \subset A_i$?
-2
votes
0answers
19 views

Elementary word set problem [on hold]

In a class of 70 students, 6 offer economics only, 18 offer economics but not mathematics, 36 offer economics and geography, 53 offers economics,50 offers geography and 34 offers mathematics and ...
-5
votes
1answer
46 views

Union Intersection question, teacher is stuck. [on hold]

$s(B) = 2 s(A)$ , $s(B\cap A') s(A\cup B) = 32$ ; so $s(A\cap B)=?$
0
votes
2answers
28 views

Writing a set with only 0 and 1 [on hold]

I want to write a set which m number of elements and the value of the elements can either be 0 or 1. How do I write this in set notation
1
vote
4answers
36 views

Number of subsets of A∪B that contain an odd number of elements

So I have a problem which defines two sets: $A = \{1,3,5\}$ and $B = \{ 1,2,3,4\}$. The question asks for the number of subsets of $A \cup B$ that contain an odd number of elements. I know the ...
-1
votes
0answers
35 views

Common character to substitute union ∪ and intersection ∩

When writing set expressions on a computer without access to the proper symbols (∪, ∩, etc.), what non-letter symbols found on a English US keyboard are commonly used as substitutes? The three ...
0
votes
1answer
27 views

Definition of factorial function for sets

Why is the factorial function expressed in terms of $(n+1)$ for sets? $0! = 1$ $(n+1)! = (n+1) \times n! $ for all $n$ $\in\mathbb{N}$ Instead of the more "common" $0! = 1$ $n! = n \times (n-1)!$ ...
0
votes
0answers
38 views

Basic Analysis . Is there a contradiction in these two definitions of closed sets for the set $x_n=\{\dfrac {1}{n}: n \in \mathbb N\}$

Is there a contradiction in these two definitions of closed sets for the set $x_n=\{\dfrac {1}{n} : n \in \mathbb N\}$ ? I happen to have a very basic query please. Let $S$ be the set denoted by ...
0
votes
1answer
59 views

How to do a basic proof

Let $f$ be a function from $A$ to $B$; let $C$, $D \subseteq B$ such that $C\subseteq D$. Prove that $$f^{-1}(C) \subseteq f^{-1}(D)$$ Work done: $x \in f^{-1}(C)= \{x\in A \ \text{such that} \ ...
-1
votes
2answers
17 views

Cardinality of the union of infinite and countable sets

This seems evident, but I cannot come up with a reasonable proof for: Question: show that if $X$ is an infinite set and $Y$ is a countable set, then $|X \cup Y|=|X|$
0
votes
0answers
15 views

Nondecreasing sequence of a set [on hold]

Find $\lim_{x\to\infty}C_k =$ {$x:\frac1k \le x \le3-\frac1k$},$C_k =$ {$(x,y):\frac1k \le x^2+y^2 \le4-\frac1k$} where k= 1,2,3,....
0
votes
2answers
24 views

belong to and subset in the set

My question is I can't understand the difference between belong and subset. Set theory: difference between belong/contained and includes/subset? I've read this already but I didn't get it yet... ...
1
vote
1answer
13 views

If $S$ is a subset of $\mathbb R^p$, then every infinite subset of $S$ has an accumulation point in $S \implies S$ is closed

If $S$ is a subset of $\mathbb R^p$, then every infinite subset of $S$ has an accumulation point in $S \implies S$ is closed. My query is : Isn't the above statement self proving? Every infinite ...
2
votes
3answers
55 views

Is my proof by contradiction about the empty set correct?

I am trying to learn about proofs and one of the exercice in my book (Maths ABC) is about proof by contradiction. I think I understand the concept but I would like to have a feedback on the following ...
0
votes
1answer
42 views

The composition of functions and inverse of a set?

I'm a bit confused on how to do some of my discrete math work. I tried doing all of the problems, but I feel like I'm doing something wrong. If anyone could correct me, it would be greatly ...
1
vote
0answers
14 views

In what sense $\alpha \times \alpha$ is the initial segment $(0,\alpha)$ in $Ord \times Ord$?

This is from Jech's book on set theory: "We define a well ordering of the class $Ord \times Ord$ of ordinal pairs. Under this well ordering, each $\alpha \times \alpha$ is an initial segment of ...
0
votes
0answers
27 views

Induction in other algebraic structures

While reading this MSE question on real induction, as well as other articles a Google search brought me to, I suddenly became interested in this topic. Induction on the monoid of natural numbers ...
-2
votes
1answer
45 views

Show that equality is symetric, reflexive and transitive [on hold]

So I'm stuck in one of Tao's exercises regarding set theory (The first one, actually). It states the following: "Show that the definition of equality in 3.1.4 is reflective, symmetric and transitive" ...
0
votes
1answer
23 views

Set containing multiple of both x and y in a certain range

Let A be the set which contains natural numbers which are multiples of 4 in the range 200 to 12000. Let B be the set which contains natural numbers which are multiples of 100 in the range 200 to ...
-1
votes
0answers
49 views

Prove that every natural number is transitive

A set A is said to be transitive if for all $x ∈ A$, $x ⊆ A$. (a) Prove that every natural number is transitive. (b) Prove that a set A is transitive if and only if $\cup A ⊆ A$. For (a), is it ...
1
vote
2answers
47 views

When is an infinite set larger than another infinite set?

Somewhat of a basic question that I've been pondering about, suppose we have 2 finite sets $A,B$, arbitrary sets with arbitrary elements that we know nothing about, except that they are both finite. ...
3
votes
1answer
31 views

Ordinals, cardinals and how to understand $2^{\aleph_0}$ versus $2^{\omega_0}$

I have read that $2^{\omega_0}=2^{\omega}=\omega$. (in the sense that they have the same order type). On the other hand, I know that $\omega=\aleph_0$, since it is the least infinite countable ...
0
votes
2answers
33 views

Show that a set is countable

I have to show that the set $B=\{n^2 + m^2 : n, m \in\mathbb N\}$ is countable. I know that i need to find a injection or a bijection from the set $B$ to the natural numbers, but i don't know how. ...
0
votes
3answers
35 views

The set of real numbers and the set of Real valued functions are not similar (equinumerous)

We need to show that the set of real numbers and the set of Real valued functions whose domain is $\mathbb R$ are not similar (equinumerous). Let $\mathbb R$ denote the set of real numbers and $S$ ...
1
vote
1answer
22 views

How do I simplify the union and intersection of sets.

In a solution to a homework problem, it says that $P((A\cup B)\cap C')=P(A\cup B\cup C)-P(C)$. I couldn't figure out how these two are equivalent. I have tried distributing $C$ into $(A\cup B)$ but I ...
2
votes
3answers
86 views

Is $|\mathbb{R}$| = |$\mathbb{R^2}$| = … = |$\mathbb{R^\infty}$|?

I know that $|\mathbb{R}| = |\mathbb{R^2}|$ because they both contain uncountably many elements, but I find it hard to conceptually understand how $\mathbb{R}$ defined by a line is the same size as ...
1
vote
2answers
25 views

What is the complement of conditional probabilities?

I am working with a problem that uses Bayes Theorem and conditional probabilities. I have the conditional probability that a plane has an emergency locator $(E)$ given that it was discovered $(D)$ ...
1
vote
2answers
40 views

Symmetric difference proof [on hold]

How can I show that $$ ( \, A \smallsetminus B \, ) \cup ( \, B \smallsetminus A \, ) = ( \, A \cup B \, ) \smallsetminus ( \, A \cap B \, ) $$ rigorously?
0
votes
0answers
34 views

What are algorithm? Can we relate algorithm using set theory concepts?

What really algorithm are? Can we define algorithm as functions or in terms of set theory (I think it is foolish what am I writing) But can we reconvert proof using algorithm in set theory concept.For ...
0
votes
1answer
21 views

Define a new operation and prove the field axiom hold for it.

Define $\ a\triangle b=ab$, where $a,b\in\Bbb R^{+}$, the set of positive real numbers. Show that $\exists x \in\Bbb R^{+}$ s.t $a\triangle x=0$. I think the statement is false, because you can not ...
1
vote
0answers
35 views

Existence of extended real number system

How can we prove the existence of Extended real no. such that order and addition and multiplication (which are also relation means set) are also extended from real no. system. Can you please provide ...
0
votes
2answers
23 views

Give the definition of S $\subseteq$ T for general sets S and T.

The answer I can come up with is; S is a subset of T, denoted by S $\subseteq$ T, or equivalently, T is a superset of S, denoted by T $\supseteq$ S. Can someone correct me if I am wrong, or provide ...
2
votes
2answers
42 views

Equation with sets

So, I got this system of equations with sets (I need to find X): $ \begin{cases} (A \cup X) \cup C = B \cap C \\ X \setminus A = B^\complement \end{cases}$ I got, that $ \begin{cases} A \subseteq ...
0
votes
1answer
36 views

If $B$ is an uncountable set and $A$ is a countable set, then prove that $B$ is similar to $B-A$.

If $B$ is an uncountable set and $A$ is a countable set, then prove that $B$ is similar to $B-A$. Attempt: Two sets $A$ and $B$ are called similar $\iff$ thee exists a one to one function $F$ ...
0
votes
1answer
18 views

Does (f(0)=g(0) or f(1)=g(1)) define a transitive relation on function?

I need is to check if a relation is an equivalence or not. I can see that it is reflexive and symmetric but I'm not able to find out if it is transitive. The relation is defined on the set of all ...
0
votes
1answer
34 views

Are any two uncountable sets similar to each other?

Two sets $A$ and $B$ are called similar $\iff$ thee exists a one to one function $F$ whose domain is the set $A$ and whose range is the set $B$. We know that two countably infinite sets should be ...
2
votes
2answers
26 views

Proving sets implication using the method of contradiction

Suppose S and T are sets. Consider the following implication: If $A∩B=∅$ and $A ∪B = B$, then $A = ∅$. Prove the given implication by contradiction. So I have started by coming up with the negation: ...
0
votes
1answer
32 views

The number of $p$-subsets of an $n$-set is $n$ choose $p$

I want to show that the number of subsets of cardinality $p$ of a set $E$ of cardinality $n$ is ${n \choose p}$. I've read a proof that I couldn't understand it basically says that for any injection ...
2
votes
2answers
61 views

A standard Operations on sets.

I am trying to show the following property of sets: $$ A \cap ( \bigcup A_{\alpha} ) = \bigcup (A \cap A_{\alpha}) $$ My attempt: Let $x \in A \cap ( \bigcup A_{\alpha} )$. This occurs iff $x \in ...
0
votes
1answer
25 views

Maximum and Minimum in Set Theory

In a group of 100 students, each student has to opt for one or more of the three subjects among Physics, Chemistry and Mathematics. The number of students who opted for Mathematics is more than the ...
0
votes
1answer
25 views

Could someone explain this conditional probability problem?

This is an example in my textbook but I do not understand it. A news magazine publishes three columns entitled “Art” ($A$), “Books” ($B$), and “Cinema” ($C$). Reading habits of a randomly selected ...
1
vote
2answers
23 views

Cardinality of a line and a half plane

intuitively it seems like the cardinality of the set of points that make up a line should be different than the cardinality of the set of points that make up a half plane but I couldn't come up with a ...
1
vote
1answer
24 views

Infinite sets: $|a| < |b|$ implies $|c^a| < |c^b|$

If $a,b,c$ are infinite sets, is it true that $|a| < |b|$ implies $|c^a| < |c^b|$? Obviously $|a| < |b|$ implies $|c^a| \leq |c^b|$, but I want to show $c^a$ does not biject with $c^b$...
0
votes
1answer
23 views

proof of set operation discrete math

if $$a=\{3n \mid n \in \mathbb{Z}^+\}$$ and $$b=\{3^{2m}\mid m \in \mathbb{Z}^+\},$$ prove that $b$ is a subset of $a$. I think the question is wrong. I think $a$ should be a subset of $b$.
0
votes
1answer
46 views

Infinite Set confusion

Given an infinite set $A$, I want to show that there exists some subset $B$ of $A$ such that $|A| = |B|$. This is the definition of an infinite set. I can create examples of this, but I am confused ...