# Tagged Questions

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### Compositions of filters on finite unions of Cartesian products

Let $\Gamma$ be the lattice of all finite unions of Cartesian products $A\times B$ of two arbitrary sets $A,B\subseteq U$ for some set $U$. See this note for other equivalent ways to describe the set ...
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### Shortest possible proof of a simple theorem [closed]

I want the shortest proof of the following not so hard theorem. I have not yet proved it, I am going to prove it. But if your proof is shorter than mine, you would win :-) Let $A$, $B$ be sets. I ...
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### Union of Cartesian squares

I am trying to find sufficient and necessary conditions for a relation to be representable as a union of Cartesian squares: $\bigcup_{i\in I}(X_i\times X_i)$ for some family $X_i$ of sets. One ...
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### Basic Cartesian prodcuts

I am having some issues grasping basic ideas of Cartesian products. I am reading a PDF my professor gave us explain sets/Cartesian products. If $\mathbb{R}\times \mathbb{R}$ can be written as ...
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### Product of Summations for All Subsets

We have a set $X$ of $n$ integers $\{$$x_1, x_2, .. , x_n$$\}$, for which there are $2^n$ total subsets. The summation $s$ of a subset $X'$ is simply the sum of all integers present in $X'$, ...
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### Can we express these sets as Cartesian products of two subsets of $\mathbf{R}$?

Let sets $A$ and $B$ be given as follows: $$A := \{ (x,y) \in \mathbf{R}^2 | \ \ x < y \ \ \}$$ and $$B := \{ (x,y) \in \mathbf{R}^2 |\ \ x^2 + y^2 < 1 \ \ \}.$$ Can we express $A$ or $B$ as ...
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### About binary relations under certain conditions and their composition

(I have edited it. The previous version was with errors.) Let $A$ be a set. Let $\pi_0$, $\pi_1$ be projections from $A\times A$. Let $F_0$, $F_1$, $G_0$, $G_1$ be binary relations on $A$. Let ...
(1) If $X$ and $Y$ are two sets, we define the Cartesian product $X \times Y$ as the set of ordered pairs $(x,y)$, such that $x \in X$ and $y \in Y$. (2) On the other hand [Folland, Real Analysis, ...