8
votes
3answers
183 views

Existence of a certain subset of $\mathbb{R}$

To every real $x$ assign a finite set $\mathcal{A}(x)\subset \mathbb{R}$ where $x\not\in \mathcal{A}(x)$. Does there exist $\mathcal{W}\subset \mathbb{R}$ such that: $$1.\;\;\mathcal{W}\cap ...
4
votes
3answers
64 views

Elementary set theory problem - I get an incorrect result

The problem given is this: $\bigcap_{i \in I}(A_i \cup B_i)$ and $(\bigcap_{i \in I}A_i) \cup(\bigcap_{i \in I}B_i)$ I am asked if they are the same. Here is the reasoning I used: for the first ...
2
votes
2answers
214 views

how to prove $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$

I am given this equation: $f^{-1}(B_1 \cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)$ I want to prove it: what i did is I take any $a \in f^{-1}(B_1 \cap B_2)$, then there is $b \in (B_1 \cap B_2)$ so ...
1
vote
1answer
55 views

elementary neighborhood problem

I am to find the proper number from $x \in \{2,3,4\}$ for which this following set is a neighborhood in $\mathbb{R}$ or in $\mathbb{C}$, $$A:= \left] 1,4 \right[ \cap \left[ 2,5 \right]$$ Firstly, I ...
1
vote
5answers
150 views

How to show $A-B \subseteq C \Rightarrow A\cup B \subseteq B\cup C$?

I really need help with this logical proof. Show that $A-B \subseteq C \Rightarrow A\cup B \subseteq B\cup C$. Please show the steps to the solution. Thank you!
1
vote
3answers
134 views

minimum of this simple set

i need again some help here. i am defining the minimum and max and inf and sup of this set $A:=(]1,2[ \cup ]2,3]) \cup \{2\}$ which is equal to the interval $(1,3]$ i say, max is 3, and sup is also ...
2
votes
3answers
451 views

Maps - question about $f(A \cup B)=f(A) \cup f(B)$ and $ f(A \cap B)=f(A) \cap f(B)$

I am struggling to prove this map statement on sets. The statement is: Let $f:X \rightarrow Y$ be a map. i) $\forall_{A,B \subset X}: f(A \cup B)=f(A) \cup f(B)$ ii) $\forall_{A,B \subset X}: ...
0
votes
1answer
233 views

Uniqueness of minimum and maximum of set

i am stuck in this simple but foggy problem. i need to prove or show that the min and sup are unique if they exist. $A$ is a nonempty set and $B$ is nonempty subset of $A$. i am trying to show that ...
0
votes
1answer
42 views

monotoneness - prove the elementary statement

Let $A, B, C$ be nonempty sets with total order. and let $f\colon A \rightarrow B$ and $g\colon B \rightarrow C$ be maps. Prove these statements: a) If $f, g$ are antitone, then $g \circ f$ is ...
0
votes
2answers
253 views

Counting functions

How many functions are possible from the set $A=\{0,1,2\}$ into the set $B =\{0,1,2,3,4,5,6,7\}$ such that $f(i) \le f(j)$ for $i \lt j$ and $i,j \in A$? I am not sure which counting model ...
2
votes
1answer
89 views

Is this relation transitive if $n=m$?

If $X$ is a set and $n \in \mathbb N$, then $[X]^n$ will denote the set of all subsets of $X$ with exactly $n$ elements. For a set $X$ and natural numbers $n$ and $m$ define a relation $R$ on $[X]^n$ ...