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Let two sets $\mathcal{A} \subset \mathbb{R}^2 = \left \{ (x_1,x_2) | x_1,x_2 \in \mathbb{R} \right \} $ and $\mathcal{B} \subset \mathbb{R}^3 = \left \{ (x_1,x_2,x_3) | x_1,x_2,x_3 \in \mathbb{R} \right \}$

How can I mathematically write that $\mathcal{B}$ is an extended version of $\mathcal{A}$ with the same number of elements?

For example if $\mathcal{A} = \left \{ (1,2) , (2,5) , (4,1) \right \}$ and $\mathcal{B} = \left \{ (1,2,1) , (2,5,0) , (4,1,3) \right \}$, for each element of $\mathcal{B}$ there is one and only one element in $\mathcal{A}$ with the same $x_1$ and $x_2$, and $\left | \mathcal{A} \right |=\left |\mathcal{B}\right |$.

Asaf Karagila
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Cristi
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    Personally, I would write "$p_{{1,2}} : \mathcal{B} \to \mathcal{A}$ is a bijection", but as this is a generalization of the notation for the canonical projections that I personally haven't seen used, I would also have to define what $p_\alpha : \prod_{i \in \mathcal{I}} X_i \to \prod_{i\in \alpha} X_i$ means for $\alpha \subseteq \mathcal{I}$. – Brian Moehring Jul 08 '23 at 17:46
  • @BrianMoehring Projections in relational algebra can have similar notation. If your tuples are all cartesian products of a set $X$, I'll just point out that every function $r:[n]\to[m]$ (with $[k]={1,2,\dots,k}$) functorially induces a "projection" $r^:X^m\to X^n$ (it's also a way to reindex as well), and you could define $p_{1,2}$ to correspond to $r^$ for $r:[2]\mapsto [3]$ with $r(1)=1$ and $r(2)=2$. That's all to say the notation makes sense with the subscript being a sequence rather than a set. – Kyle Miller Jul 10 '23 at 08:39
  • With this notation, the standard projections $p_i:X^n\to X$ are each the $r^*$ where $r:[1]\to [n]$ is defined by $r(1)=i$. It's kind of neat that the generalization generalizes the notation too. – Kyle Miller Jul 10 '23 at 08:41

2 Answers2

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I would do it saying that the function $$f:B\to A$$ given by $f(x,y,z)=(x,y)$, is a bijection.

Jackozee Hakkiuz
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You may say $B$ is a graph of a real-valued function with domain $A$ in $\mathbb R^2$ – but I don't know if there exists any mathematical symbol (notation) for it.

CiaPan
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