Given two sets of objects:
- $a = (x, y, z)$ where $x \in X$ and $y \in Q \cup R$ and $z \in P \cup T$
- $b_1 = \{ x \mid x \in X \}$ and $b_2 = \{ y \mid y \in Q \cup R \}$ and $b_3 = \{ z \mid z \in P \cup T \}$
How to go about "proving" that they are different aspects of the same object.
I have never done proofs before and am trying to come up with a practical example outside of the traditional logic examples provided in introductory material. These two sets of objects $\{a\}$ and $\{b_1,b_2,b_3\}$ both define the same thing but in different ways. I'm wondering if there is some sort of mapping I can do to show for sure that in fact these objects are the same. Not sure how to go about this. For instance, in programming these two objects can be shown the same by essentially merging the x, y, z components of $b_1,b_2,b_3$ into $(x, y, z)$, then iterating through each item in the set and comparing the "where" clauses between the new $b$ and the existing $a$. If done with enough detail, there would be a direct mapping between the two "forms" of the same object, which seems like it is close to a proof. So I am wondering how to do this in a mathematical way.
Your confusion may come from a notational problem, so let me rewrite your definition in a formal way. To avoid any ambiguity, I will use capital letters for the sets and lower case letters for the elements. Now, my understanding of your definitions is the following: $$ B_1 = \{x \mid x \in X\} = X,\quad B_2 = \{ y \mid y \in Q \cup R \} = Q \cup R,\quad B_3 = \{ z \mid z \in P \cup T \} = P \cup T. $$ $$ A = \{(x,y,z) \mid x \in X, y \in Q \cup R, z \in P \cup T\} $$ Therefore, you could alternatively write $A$ as follows $$ A = X \times (Q \cup R) \times (P \cup T) = B_1 \times B_2 \times B_3 = \{(b_1, b_2, b_3) \mid b_1 \in B_1, b_2 \in B_2, b_3 \in B_3 \} $$