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, de Morgan's laws, Venn diagrams, relations, functions, ...

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1
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1answer
60 views

Is this definition valid?

I am working on this problem: "Suppose $f:A\times A\rightarrow A$. A set $C \subseteq A$ is closed under $f$ if $\forall (x,y) \in C \times C(f(x,y) \in C)$. Now suppose $B \subseteq C $. The closure ...
0
votes
1answer
22 views

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$.

Let $(X,\mathfrak T)$ be a topological space and suppose that A and B are subsets of X such that $A \subseteq B$ then $Cl(A) \subseteq Cl(B)$. My definition of closure is "Let $(X, \mathfrak T)$ be ...
3
votes
1answer
23 views

How do I find the coordinate relationship between numbers on a number spiral?

For instance, considering the number spiral below, If I wanted to say where the number $10$ was in relation to the number $18$, I might say something like relationship $(18,10) = (4,-2)$ since it is ...
0
votes
2answers
30 views

Have I Correctly Defined the Set of Nonzero Complex Numbers $\mathbb{C^*}$?

If the set of complex numbers $\mathbb{C} = \{a+bi\mid a,b \in \mathbb{R}\}$, then what would be the definition of the set of nonzero complex numbers? Am I right in defining such a set as ...
3
votes
2answers
36 views

Set theory (containing Power Set) Need Help in a proof

I am confirming whether my proof is correct or not and need help. If $ A \subseteq 2^A , $ then $ 2^A \subseteq 2^{2^A} $ Proof: Given: $ \forall x ($ $ x\in A \rightarrow \exists S $ where $ ...
2
votes
1answer
17 views

The domain of a function as a function: the “domain-function”

The domain of a function $f:X\to Y$ is normally defined as $\operatorname{dom}f\equiv X$, but I would like the domain-function $\operatorname{dom}$ to be a funtion itself, i.e. I would like to define ...
-2
votes
3answers
30 views

sets union problem [closed]

If $A$ and $B$ be two sets containing 3 and 6 elements, what can be minimum and maximum number of elements in $A\cup B$.
0
votes
1answer
18 views

Let $(X, \mathfrak T)$ be topological space and suppose that A and B are subsets of X such that $A \subsetneq B$. Then $Int(A) \subsetneq Int(B)$.

Let $(X, \mathfrak T)$ be topological space and suppose that A and B are subsets of X such that $A \subsetneq B$. Then $Int(A) \subsetneq Int(B)$. ( $\subsetneq$ means "is a proper subset") My ...
0
votes
0answers
21 views

Prove that h:$\mathbb{Z}\rightarrow\mathbb{O}$ where h(n)=2n-1 is bijective

I need to prove that h:$\mathbb{Z}\rightarrow\mathbb{O}$ where h(n)=2n-1 is bijective. I haven't done problems where $\mathbb{Z}\rightarrow\mathbb{O}$ and have seen no examples. I am only familiar ...
0
votes
1answer
49 views

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. If $Bd(A) = \emptyset$ then A =∅ or $A = X$ .

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. If $Bd(A) = \emptyset$ then $A = \emptyset$ or $A = X$. I am studying introduction to proofs and we have learned ...
3
votes
2answers
106 views

Does set difference distribute over set intersection?

I am asked to prove that, if $A, B$ and $C$ are sets, then $$A-(B\cap C)=(A-B)\cap(A-C).$$ However, I think that either I have made an error in my working, or the wording of the problem contains a ...
0
votes
1answer
18 views

monotonic laws for ordinal subtraction

I have to prove some monotonic laws for ordinals. It's quite comfortable for me to show monotonic laws of ordinal addition (e.g. $\beta\leq\gamma\Rightarrow\alpha+\beta\leq\alpha+\gamma$). But when it ...
3
votes
5answers
52 views

Sets $A,B,C$ with $B\subseteq C$, prove that $(A-B)-C=A-C$

Ran across this and couldn't figure out how you would give a formal proof. It seems intuitive, in that $(A-B)-C$ is the elements in $A$ but not in $B$, and then also remove the elements from $(A-B)$ ...
2
votes
0answers
23 views

Hasse Diagram Correct?

I've had to make Hasse Diagrams before, but they've always been, for lack of a better word, pretty. The lines haven't had any complicated back and forth or the like. The jump that 4 and 6 have to do ...
2
votes
4answers
60 views

Are there uncountably many subintervals in $[0,1]$?

Are there an uncountably infinite number of sub-intervals in $[0,1]$ such that the number of real numbers in each of those sub-intervals is uncountably infinite? I would say no, because you would ...
1
vote
3answers
86 views

Is this a valid logical paradox?

In some recent cases, I have noticed some theorems are accepted to be intuitively or logically true if they themselves, as a unit, have no valid proof, but, their statements can be used to prove ...
-2
votes
1answer
69 views

Set Theory and $\frac{1}{3} = 0.333\dots$ [duplicate]

If we have a set like this $\{\,0.3, 0.33, 0.333, \dots\}$ Then, will the number $\frac{1}{3}$ or $\ 0.333\dots$ be a member of this set? My notation is synonymous to this notation. ...
0
votes
2answers
14 views

Lattice orders and number of elements in a set

My discrete mathematics lecture notes give the following definition of a lattice order: A 'Partial order R is a lattice order if the set of lower bounds for any two elements $x, y ∈ X$ has the ...
1
vote
2answers
41 views

$| A\cap B| = |A \cup B|$ and $A$ is different from $B \implies A \cup B$ infinite

If the power of $|A\cap B|$ equals to the power of $|A \cup B|$ and the sets $A,B$ are different, $\implies A \cup B$ is infinite. How can I prove this?
2
votes
1answer
20 views

Prove that $A\subseteq B$ if and only if $A^{C}\cup B=\mathscr{U}$

Prove that $A\subseteq B$ if and only if $A^{C}\cup B=\mathscr{U}$. I know we have to show that: if $A\subseteq B$ then $A^{C}\cup B=\mathscr{U}$ if $A^{C}\cup B=\mathscr{U}$ then $A\subseteq ...
11
votes
3answers
404 views

Why is the collection of all algebraic extensions of F not a set?

When proving that every field has an algebraic closure, you have to be careful. In this article https://proofwiki.org/wiki/Field_has_Algebraic_Closure, and as I have been told on this site, if we have ...
1
vote
1answer
31 views

Help understanding a proof about cardinal numbers

I was reading a proof about cardinal numbers, but I do not understand one step. The proof goes as follows: "Let $\beta$ be any ordinal, and for each ordinal $\alpha \lt \beta$, let $\kappa_{\alpha}, ...
0
votes
2answers
33 views

Prove a set is a subset of another. [duplicate]

I need to prove $A⊆B$ where A and B are defined as: ${A =\{x | x = 2n + 1}\}$ ${B =\{x | x = 2m - 21}\}$ where $n,m∈\mathbb{Z}$ I know that I need take an arbitrary element from A and show that it ...
0
votes
1answer
36 views

If $A$, $B$, $C$ are any infinite sets then is $|A|=|B|$ and $|A|=|C|$ $\Longleftrightarrow |A|=|B\cup{}C|$?

Suppose we have three sets $A$, $B$, and $C$ that we know are infinite sets, but we do not know anything else about the cardinality of $A$, $B$, and $C$. Is $|A|=|B|$ and $|A|=|C|$ ...
0
votes
0answers
12 views

Proof the inverse image of set difference

I've the following exercise: Let $f:A \to B$ with $C,D \subseteq B$. Prove that $f^{-1}(D-C)=f^{-1}(D)-f^{-1}(C)$ For the proof, I've started from the definition of subsets a) $C,D \subseteq B ...
2
votes
2answers
101 views

Let $(X ,\mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then $Cl(A) = A\cup Bd(A)$. False!

Let $(X ,\mathfrak T)$ be a topological space and suppose that $A$ is a subset of $X$. Then $Cl(A) = A\cup Bd(A)$. I think this statement is false because the definition of closure does have the ...
2
votes
4answers
129 views

$4$-element subsets of the set $\{1,2,3,\ldots,10\}$ that do not contain any pair of consecutive numbers

Find the number of subsets of the set $\{1,2,3,\ldots,10\}$ that contain $4$ elements and do not include any pair of consecutive numbers. For example $\{1,2,5,7\}$ is not an example of such a subset ...
2
votes
2answers
40 views

Set Theory Proof: Valid or not?

I'm trying to gain understanding of set proofs and I came across this one. I can't help but think the proof is too simple and that there is more to it. Problem: Prove or disprove for arbitrary sets ...
0
votes
1answer
8 views

To proof the difference images is a subset of their map difference sets

Let $A$ and $B$ sets, with $P,Q \subseteq A$ and let $f:A \to B$ 1) prove that $f(P)-f(Q) \subseteq f(P-Q)$ 2)Is it necessarily the case that $f(P-Q) \subseteq f(P)-f(Q)$? Give a proof or a ...
1
vote
2answers
22 views

Finding the cardinality of a cartesian product of a set and a cartesian product.

$A = \{0, 1, \{2, 3, 4\}\}$ $B = \{1, 5\}$ $C = B \times \mathbb{N}$ What is the cardinality of $A \times C$? I know the enumeration of $A \times C$ is $\{(0,(1,0)), (0,(1,1)), (0,(1,2))\ldots ...
0
votes
1answer
58 views

Proper notation for the function $g(x) = x^2+6$.

I'm using this more as a method of verifying if I'm correct on a question I am having difficulty with. Keep in mind, I'm a complete beginner, so.. yeah. Thereom: Assume the function $g$ is ...
0
votes
0answers
12 views

what can you say about this operation using principle of inclusion/ exlusion [duplicate]

Given: A = [ Aaron features ], B = [Bob features], X = [all countries in Europe ] What must be true if: |A ∩ B ∩ X| = 1 Even though I don't see the relation between cardinality of sets, I kind ...
1
vote
1answer
27 views

Difference between Zorn's Lemma and the ascending chain condition

Let $S$ be a non-empty partially ordered set with respect to a relation $\leq$. Then: Zorn's Lemma: If $S$ has the property that any totally ordered subset $U\subset S$ has an upper bound, then ...
0
votes
2answers
35 views

Proof or a counterexample of a function

I have the following exercise, how can I proceed? Let $A$ and $B$ be sets, with $S \subset A$ and $f:A\to B$ a function, and $g:A\to B$ be an extension of $f\rvert_S$ to $A$. Does $g$ equal $f$? ...
0
votes
3answers
51 views

The number of sets $X$, such that $X\subseteq A$ and $X \subsetneq B$, where $A=${$a,b,c,d,e$}, $B=${$c,d$}

The number of sets $X$, such that $X\subseteq A$ and $X \subsetneq B$, where $A=${$a,b,c,d,e$}, $B=${$c,d$}(1)26(2)27(3)28(4)29 Answer: (3) well hard to say but i don't know what is actually ...
-1
votes
1answer
20 views

Let A be a countable set of points on the plane. Prove that the remaining part of the plane is connected. [closed]

Let A be a countable set of points on the plane. Prove that the remaining part of the plane is connected : any two points outside A can be connected by a polygonal line (with two segments) that does ...
4
votes
2answers
55 views

Cardinality of a collection of subsets of R

Suppose A is a family of subsets of R with the property that the intersection of any two sets in A is finite. Show that $|A|\leq 2^{\aleph_0}$. I was told that choosing a countable $D \subset B$ for ...
2
votes
1answer
47 views

Where does this function come from in this proof?

This is an excerpt taken from a proof: Let each $M_n(n\in\mathbb{N})$ be countable, Then there exists an injective function $f_n:M_n\rightarrow\mathbb{N}$. Now, set a function ...
1
vote
1answer
42 views

compute the number of subset of {1,2,3,4}

How many triple $(A,B,C)$ of subsets of $\{1,2,3,4\}$ satisfy to the following condition? $$(A\cap B)\subseteq C \subseteq (A\cup B)$$ I think every element of $\{1,2,3,4\}$ has 4 choice ,or in $A$ ...
1
vote
1answer
32 views

How do you define a set of all numbers with $N$-length decimal expansions

That is to mean a set of all numbers with $0,1,2,\dots,N$ decimal places. If I was going to generate the set I would just use a step of the $N$th unit, so if $N = 5$ then my step would be $0.00001$. ...
1
vote
1answer
30 views

How to represent a Neither/ Nor set operation

Given: A = { Aaron friends } B = {Bob friends} X = {all members in network} Set of friends in the network that are friends of neither Aaron nor Bob I got as answer: $X-(A \cup B)$ Is this ...
0
votes
1answer
15 views

Cartesian Product of a normal set and a set with an element which is a set.

What is the Cartesian product of A and C? Where: A = {0, 1, {2, 3, 4}} C = {1, 5} Pretty standard for the first 2 elements of A. So far I have {(0,1), (0,5), (1,1), (1,5).... } But for the last ...
5
votes
4answers
49 views

Using disjunction to prove that $A \setminus (A \setminus B) = A \cap B$

The problem is as follows: Suppose $A$ and $B$ are sets. Prove that $A \backslash (A \backslash B) = A \cap B$. I've rewritten the problem as a biconditional where $A \backslash (A \backslash B) ...
1
vote
0answers
30 views

Is the boundary of a set a subset of the limit points?

Let $(X, \mathfrak T)$ be topological space and suppose that $A$ is a subset of $X$. Then $Bd(A) \subseteq A'$. My definition of boundary: Let $(X,\mathfrak T)$ be a topological space and let $A ...
0
votes
4answers
33 views

The point 2 is not a limit point of the set $[0,1)$ regardless of the topology on $\mathbb R$.

The point 2 is not a limit point of the set $[0,1)$ regardless of the topology on $\mathbb R$. Is this true or false? I think it is true because the limit point has to be a subset of the set and 2 is ...
-1
votes
2answers
25 views

Prove the $Int(A) \subseteq A$. Using elements/ sets

Prove the $Int(A) \subseteq A$. My definition of interior is Let $(X, \mathfrak T)$ be a topological space and let $A \subset X$ is the set of all points $x \in X$ for which there exists an open set ...
0
votes
1answer
32 views

If A is a subset of a topological space, then $Bd(A) \subseteq Cl(A)$. Prove using elements/ sets.

If A is a subset of a topological space, then $Bd(A) \subseteq Cl(A)$. Prove. I know this statement is true. I am now trying to prove it. I am in a basic topology class and to do a lot of set ...
0
votes
1answer
30 views

If $A$ is a subset of a topological space, then $A' \subseteq A$ versus For any closed subset $A$ of a topological space, $A' \subseteq A$.

I need to determine which of the following are true and prove it... if it is false then I have to give a counterexample. If $A$ is a subset of a topological space, then $A' \subseteq A$ versus ...
0
votes
1answer
15 views

Set notation and translation

I've been checking my understanding of sets by setting up the following for myself: $\{ 1, 3, 5, 7, ... \}$ = the set of all natural odd numbers $\{ ..., -4, -2, 0, 2, 4, ... \}$ = the set of all ...
-1
votes
0answers
21 views

For all A∈F, prove uF=Ø if and only if A=Ø

The title contains the proof my friends and I are trying to solve for our class. We think we have it the forwards way, proving uF=Ø iff A=Ø, but we are having trouble proving it in reverse