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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0
votes
1answer
14 views

Proving that $B_1$ and $B_2$ doesn't have maximal element

This is one of the problem I have been solving form Velleman's How to prove book: Suppose $R$ is a partial order on $A$, $B_1 \subseteq A$, $B_2 \subseteq A$, $\forall x \in B_1 \exists y \in B_2 ...
1
vote
1answer
19 views

Proof set theory involving instantiation

Is it okay to instantiate with the same element in universal and existential instantiation? Here follows my proof of the following theorem. Theorem If $A \subseteq B \setminus C $ and $A \not = ...
1
vote
0answers
34 views

Set Theory Proof for Trignometry

How do I solve whether $A$ is a subset of $B$, $B$ a subset of $A$, and whether either are proper subsets of each other for the following $$A=\{x \in \mathbb{R} \mid \cos x \in \mathbb{Z}\},\\ B=\{x ...
1
vote
2answers
29 views

$A\subseteq B\to C\setminus B\subseteq C\setminus A\,$ — how to prove this?

Given $A \subseteq B $. Prove for every set $C, C\setminus B \subseteq C \setminus A $. Logical Argument: Given: $\forall x, x \in A \rightarrow x \in B $ Goal: $\forall C \forall x , x\in ...
1
vote
0answers
23 views

A subset b if and only if complement b subset complement a?

hi already tried to prove it in my own way i would like to share my results hoping a mathematician somewhere could tell if I am right or wrong I think actually that the proof consists of 2 parts First ...
0
votes
1answer
35 views

Let $A = \{1- \frac 1n : n \in \mathbb Z ^+\}$ is closed under certain topologies on $\mathbb R$.

Let $A = \{1 - \frac 1n : n \in \mathbb Z ^+\}$ is closed under certain topologies on $\mathbb R$. I am supposed to figure out if this set is closed under certain topologies. I know that means I ...
0
votes
2answers
13 views

Suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$.

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of $X$ such that $A \subseteq B$ then $Int(A) \subseteq Int(B)$. I know this a true statement so now I need to ...
2
votes
1answer
24 views

Regularity of $\omega_1$ and axiom of choice

Why is the regularity of the ordinal $\omega_1$ a consequence of the axiom of choice?
3
votes
1answer
38 views

Some unique representation of nonnegative integers

Let $\mathbb N$ be the set of nonnegative integers, that is $\mathbb N=\{0,1,2,3,\ldots\}$. Does there exist a subset $K\subset\mathbb N$ such that every $n\in\mathbb N$ has a unique ...
-3
votes
2answers
56 views

I can not prove it I can not show its proof [on hold]

Let A and B be finite sets.Prove that $A \cup B$ is a finite set and that $|A\cup B|=|A|+|B|-|A \cap B|$
12
votes
8answers
912 views

Can we have a one-one function from [0,1] to the set of irrational numbers?

Since both of them are uncountable sets, we should be able to construct such a map. Am I correct? If so, then what is the map?
1
vote
1answer
27 views

How to understand this definition of equivalence relations

I often see this type of definitions of equivalence: Suppose that $f$ and $g$ are differentiable on $\mathbb R$. We can define an equivalence relation on such functions by letting $f(x) \sim g(x)$ ...
3
votes
1answer
26 views

Defining Equivalence relations

So I am not really comfortable with equivalence relations, so this example from Wikipedia gives me trouble. Here is what it says: Let the set $\{a,b,c\}$ have the equivalence relation ...
1
vote
1answer
45 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
18 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
22 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
28 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 ...
2
votes
1answer
28 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
14 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
26 views

sets union problem [on hold]

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
13 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
20 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
45 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
92 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
9 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
46 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
21 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
57 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
76 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
50 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
381 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 ...
0
votes
1answer
11 views

Proof that the image of a sets family union is a subset of union of images [on hold]

Let D be a family of subsets of A. Prove that g(TX∈D X)⊆TX∈D g(X). As i can to proceed for prove it?
1
vote
1answer
28 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
31 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
35 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
10 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
75 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
118 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
39 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
7 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 ...
0
votes
1answer
25 views

How to prove the two functions are inverse to each other? [on hold]

How to prove the definition of following inverse function: Let A and B be two sets. Prove that: $f^{-1} : B \rightarrow A $ is an inverse function of $f : A \rightarrow B$ . (in general)
1
vote
2answers
15 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 ...
0
votes
1answer
48 views

Is the following statement true or false? [on hold]

Let $X \subseteq Y \subseteq Z$. Then $(Z \setminus Y) \subseteq (Z \setminus X)$. I am studying to be an electronic engineer, and am learning set theory and fundamentals in abstract mathematics ...
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
33 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 $f$? ...