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Let $A = \{(a_1, b_1), (a_2,b_2), \ldots, (a_n,b_n)\}$ be a set. $a,b$ may be numbers, generic objects.

Take the set $A_a = \{a_1, a_2, \ldots, a_n\}$

Does there exist any formal relationship between $A_a$ and $A$?

For instance, is $A_a$ a subset of $A$?

Is $A_a$ obtained by $A \backslash A_b$, where $A_b$ is analogously defined?

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  • $\begingroup$ $A_a$ is most certainly not a subset (they have NO elements in common!). $\phi: A\to A_a$ via $\phi;(a_n,b_n) \mapsto a_n$ is well-defined function. $A$ is a s set of $2$-tuples (aka ordered pairs) and $\phi$ is called a "projection"; it maps a set of ordered pairs to their corresponding first term. $\endgroup$
    – fleablood
    Commented Sep 12, 2020 at 5:08
  • $\begingroup$ Actually I think the standard notation of $A= \{\textbf{x}_\alpha\}\subset X_1\times X_2 \times ...\times X_n$ is set of ordered $n$-tuples where each $\textbf{x}_\alpha = (x_{\alpha_1},x_{\alpha_2},....., x_{\alpha_n})$ is an orderded $n$-tuple. Then $\pi_k: A \to X_k$ is the function for the projection that maps $\pi_k: \textbf{x}=(x_{1},x_{2},....., x_{n})\mapsto x_{k}$. $\endgroup$
    – fleablood
    Commented Sep 12, 2020 at 5:17

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Definitely, not a subset since the elements of $A_a$ do not belong to $A$. You can think of $A$ as a subset of the Cartesian product of two sets $A_1$ and $A_2$, where the $a$'s and the $b$'s belong. Then $A_a$ would be the projection of $A$ onto the first factor of the product.

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  • $\begingroup$ Good ole projection. Thanks! $\endgroup$
    – Olórin
    Commented Sep 12, 2020 at 5:07

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