# Questions tagged [elementary-set-theory]

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, differences and complements, De Morgan's laws, Venn diagrams, relations and functions, countability and uncountability, etc. More advanced topics should use (set-theory) instead.

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### Parenthesis needed? in $(S\subset T)\vee(S=T)\iff S\subseteq T$ [closed]

In the statement $$(S\subset T)\vee(S=T)\iff S\subseteq T$$ are the parenthesis formally incorrect correct but deemed best removed correct and optional correct and deemed required for readability ...
240 views

### Is $\{1,2,3\}$ equal to $\{\{1,2\},3\}$ since $\{1,2\}$ is a subset of $\{1,2,3\}$?

I have a problem with subsets in Set Theory at its most basic level. a set $A$ is a subset of set B if all the elements of $A$ are also elements of $B$. $\{1,2\}$ is a subset of $\{1,2,3\}$ Simple ...
28 views

### Power set $\mathcal{P}(A)$ with the $\subseteq$ relation is a partial ordered set, however it may not be a total ordered set

I have a question that i cannot understand. Question Let $A$ be a set. Show that the power set $\mathcal{P}(A)$ with the $\subseteq$ relation is a partial ordered set, however it may not be a total ...
43 views

### If the product topology on $\mathbb{R}^2$ is all sets of the form $(a,b)\times(c,d)\subseteq\mathbb{R}^2$ what is product topology on $\mathbb{R}^4$?

Probably the answer to this question is too obvious for it to be asked but I want to make sure my assumption is correct. Consider the real line $\mathbb{R}$ with the topology that has basis consisting ...
131 views

### $A,B$ such that $A\cap B=\emptyset$ and $A\cup B=\mathbb{R}$ and $B=\{x+y : x,y\in A\}$?

If set $A,B$ satisfy $A\cap B=\emptyset,A\cup B=I$, and $B=\{x+y : x,y\in A\}$, can $I$ be real number set $\mathbb{R}$? I think the answer is yes, but I can't construct it. If $A$ is odd number set, ...
36 views

### Prove for the ideals $I,J$ that $J\nsubseteq I$

consider the ideals $I=\langle f_1,f_2\rangle$ and $J=\langle h_1,h_2\rangle$ in $\mathbb{Q}[x,y]$. I want to prove that $J\nsubseteq I$. I'm trying to do this by proving that every element from $J$ ...
1 vote
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36 views

### can we replace two elements if they are Isomorphic to each other.

If lets say I have two sets $A =\{a_{1},a_{2},a_{3}\}$ , $B =\{b_{1},b_{2},b_{3}\}$ if the two sets A and B are Isomorphic, then can I write $a_{1}= b_{1},a_{2}=b_{2},a_{3}=b_{3}$ or can i replace A ...
77 views

### Is it meaningful to invert large cardinals?

As cardinal numbers involve the notion of ever increasing multitudes of things, is there a mathematically useful concept of ever decreasing multitudes? We already have rationals tending to zero, and ...
1 vote
44 views

### proof performed in inclusion of sets

I want to prove the following identity: $$(A\cup B)\cap (A\cup C)\subseteq A\cup (B\cap C)$$ From this equivalence I know that for a full test I should be able to prove that the left term is included ...
28 views

1 vote