Tagged Questions

3answers
76 views

Number of images from $\mathbb{N}$ to {0, 1}.

Are the number of images from $\mathbb{N}$ to {0, 1} countably infinite or uncountably infinite? I was thinking of counting in base 2 to make a bijection between $\mathbb{N}$ and {0, 1}. So, a ...
1answer
26 views

Is this relation symmetric

$R = \{(X, Y) \in \mathscr{P}(A)^2| X \subset Y \text{ and }X \neq Y \}$ I know that $(X,Y) \in R$ holds true since $X \subset Y$. However I'm unsure if $(Y,X) \in R$ since if $Y \subset X$ then ...
1answer
52 views

Is the subset relation on the powerset of a set, with qualification, reflexive?

I was wondering if the subset relation is reflexive? $R = \{(X, Y ) \in P(A)^2\mid X\subseteq Y \text{ and } X \neq Y \}$ I assumed they it was reflexive since for all $X \in P(A), X \subseteq X$ is ...
2answers
88 views

ZFC and apples described using only fundamental axioms (complete expanded reasoning)

Let's assume that I'm adding two numbers representing my count of objects I perceive (lets say a green and a blue apple that are consider to be of the same class) and I see them as a set of two apples ...
2answers
15 views

Mutually disjoint implying complements in set theory

No homework tag because it is just practice for a final, not for marks: $\text{Let$S, T \subseteq U$. If$S \bigcap T= \emptyset$, then$S$and$T$:}$ A) are always complements of each other in ...
1answer
41 views

Proper Set Theory Transformation

I was wondering if i am using the Inverse Laws Correctly in this transformation: 1. $\mathrm{A}\cup(\mathrm{B}\cap(\mathrm{A}\cup\mathrm{C})\cap(\mathrm{A}\cup\neg\mathrm{C}))$ 2. ...
1answer
59 views

Set Theory Laws

I have been working on the Inclusion Exclusion Principal and came across a problem where I am having difficulty identifying the transformation. Given Information: $\mid\mathrm{U}\mid = \mathrm{50}$ ...
2answers
77 views

Number of ways to select numbers, each 1 from different lists without repetition

I want the numbers of ways to select numbers each 1 from different lists without allowing repetition. Eg- List 1 : 5, 100, 1 List 2 : 2 List 3 : 5, 100 List 4 : 2, 5, 100 I want to select 1 ...
0answers
37 views

How to describe any partition a set

For ignore of a better word, I will use word "partition" try to describe what I mean. How to describe partition(where over lapping subsets are allowed) of a set mathematically? In another word, ...
2answers
62 views

Explanation of the formula $f^{-1}(Y)=\{x \in A |f(x) \in Y\}$ for the preimage of a set

So I found a Definition in the book that goes like this to find the pre-image of a set: $$f^{-1}(Y)=\{x \in A |f(x) \in Y\}$$ Example of the theorem being used: Let $A = \{1,2,3,4,5,6\}$ and ...
4answers
83 views

Book/Article recommendation

I am a first year Math major in the university, this summer I want to self study and go over some specific subjects. Firstly, can someone can give a suggestion for a detailed book/article about the ...
2answers
27 views

If $R$ is a transitive realation, then $R\circ R\subseteq R$

Here's the question I'm struggling with: Let R be a transitive relation on a set A. Prove the R composed with R is a subset of R. I'm kind of lost on how to prove this. I've started with saying: ...
1answer
42 views

Prove a statement with elements for Set Theory

I am stuck on this proofing question and I would like some clarification. Q: $A\subseteq B \iff A\cap B^{\prime} = \emptyset$ I already proved that LHS goes to RHS, but I am confused for the other ...
1answer
40 views

Proving a Subset Identity

Working on part A of this problem: I worked out the first part like this: 1) If $A$ is a subset of $B$ then $\forall~x~[x\in A \implies x\in B]$ 2) Same goes for $C$ being a subset of $D$ (If ...
2answers
31 views

Subsets and Cardinality

I'm confused on if I should count a subset as one element or if I should count all the elements of that subset when computing cardinality. Example: Given the set $A = \{1,2,3,\{4,5,6\}\}$ does $A$ ...
1answer
30 views

Find how many People Like dancing Only,People Like Movies

A survey was conducted among 402 persons regarding their interest in movies,dancing and games it was found that (i) 100 People Like games. (ii) 142 People Like movies or dancing but not games. (iii) ...
3answers
32 views

Using set theory to count the possible paths on an XY plane

I'm taking an introductory discrete math course, and we're studying set theory. It's going okay, but I read an example problem which gave me some difficulty. I've included a screenshot of the problem. ...
2answers
35 views

Discrete Math and Sets and subsets question

Let Universe be {1,2,3,4,5,6} If A = {1,2,3,4} then |A| = 4, and from this we can see that A is an element of U(universe), but can someone explain to me why {A} is NOT an element of U? I'snt the ...
0answers
24 views

A set system generated by a closure operator?

Given a ground set $E$, and a matroid closure operator $\tau$ on $\mathcal P(E)$, we can define a set system $(E,F)$ with $$F := \{X \in \mathcal P(E): \forall x \in X, x \notin \tau(X-\{x\}) \}$$ ...
0answers
40 views

How do we prove that, if $\mathcal{P}(A) \sim \mathcal{P}(B)$, then $A \sim B$? [duplicate]

The converse--if $\ A \sim B$ then $\mathcal{P}(A) \sim \mathcal{P}(B)$--is very easy to prove. I can't see an immediate, simple proof for the converse case. It seems like a potentially good strategy ...
1answer
36 views

Cardinality of all inverse functions (bijections) defined on: $\mathbb{R}\rightarrow \mathbb{R}$

What is the cardinality of all inverse functions defined on: $\mathbb{R}\rightarrow \mathbb{R}$? easy to calculate the upper bound which is $2^\aleph$. ($\aleph$ is the cardinality of the ...
0answers
29 views

Notations for elementary set theory

I wish to understand the following notations which defined for every cardinals, $a,b$: $P(a,b) = \left| In(A,B) \right|$ $C(a,b) = \left| P_b(A) \right|$ $E(ab) = \left| Eq(A,B) \right|$ ...
0answers
41 views

1answer
36 views

When proving a partial order relation is a total order do we have assume both elements are distinct?

Consider the "divides" relation on the set $A=\lbrace 1,2,2^2,.\;.\;.,2^n\rbrace$, where $n$ is a non-negative integer. Prove that this relation is a total order on $A$. First we prove $A$ is a ...
1answer
31 views

Equivalence Relations and distinct equivalence classes

$A=\lbrace(1,3),(2,4),(-4,-8),(3,9),(1,5),(3,6)\rbrace$. $R$ is defined on $A$ as follows: For all $(a, b)\;(c, d) \in A$, $(a, b) R (c, d) \iff ad=bc$ I know what they are asking but I cannot see ...
1answer
24 views

Solving a poset for less than equal?

I don't completely understand posets yet, so I'm confused on how to do this particular problem. Here is the question: Let S be the set of all real numbers. Prove that the less than or equal to ...
3answers
69 views

Showing a subset is uncountable [closed]

How do I show if $A \subseteq B$, and $A$ is uncountable then $B$ is uncountable?
1answer
57 views

Proving a Bound for Oddtown-Eventown or Clubtown

Suppose we have a town with a set of residents $V$, where $|V| = n$. The residents like forming clubs, and we have clubs $C_1,C_2,\ldots,C_m \subseteq V$. We are interested in the maximum number of ...