# Is $A_a = \{a_1, a_2, \ldots, a_n\}$ a subset of $A = \{(a_1, b_1), (a_2,b_2), \ldots, (a_n,b_n)\}$?

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?

• $A_a$ 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. Commented Sep 12, 2020 at 5:08
• 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 $\pi_k: \textbf{x}=(x_{1},x_{2},....., x_{n})\mapsto x_{k}$. Commented Sep 12, 2020 at 5:17

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.