Inductively show that "the ordered n-tuple $(x_1,\ldots,x_n)$ of a set so that $(x_1, \ldots,x_n) = (y_1,\ldots,y_n)$ if their coords are ordered

Question

Provide an inductive definition of the ordered n-tuple $(x_1,\ldots,x_n)$ of elements $x_1,\ldots,x_n$ of a set so that $(x_1,\ldots,x_n)$ and $(y_1,\ldots,y_n)$ are equal iff their coordinates are equal in order, i.e. $x_1=y_1,\ldots,x_n=y_n$.

My work:

To form the inductive proof, we must show that firstly $n=1$ is true, then show that $n=k$ is true, assuming that $n=k-1$ is also true.

I believe that the proof of $n=1$ is trivial, because we can show that an ordered 1-tuple set $(x_1)$ is equal to $(y_1)$ iff the coordinates are the same. However, I am finding difficult to prove the second have of the inductive proof. Any help would be greatly appreciated!!

NOTE: this question is a repost from a few days ago that was put on hold and subsequently never answered. I have edited the question sufficiently since that time and hope that we can find a solution.

Answer 1

You're asked a definition, not a proof.

Let $n$ be the number of coordinates.

1. For $1$-tuples, we define $(x_1)=(y_1)$ to mean $x_1=y_1$.

2. Now, for $(n+1)$-tuples, we define $(x_1,\dots,x_n,x_{n+1})=(y_1,\dots,y_n,y_{n+1})$ to mean $(x_1,\dots,x_n)=(y_1,\dots,y_n)$ AND $x_{n+1}=y_{n+1}$.

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Added

The question is not asking you to prove anything, you're asked to define what we mean by the expression $$ (x_1,\dots,x_n)=(y_1,\dots,y_n) $$

And you're asked even more: In given definition, we must have that (informally)

$$ (x_1,\dots,x_n)=(y_1,\dots,y_n) \iff x_1=y_1,\,\dots\,, x_n=y_n $$

Now, if you were not asked that, we could define equality of tuples as

$$ (x_1,\dots,x_n)=(y_1,\dots, y_n) \iff x_2=y_1 $$

For example (provided $n\geq 2$). This definition, of course, makes no sense if we want to use $n$-tuples as we usually do, but it is one of the possible definitions.

Answer 2

Let $X$ and $Y$ be sets. Define $X \times Y$ the set of ordered 2-tuples of $X$ to be the set $$X\times Y=\{\ \{\{a\},\{a,b\}\} \ \ | \ a\in X, b\in Y\}.$$ This set exists by the usual axioms of ZF.

Now this is a bit obscure, but the idea is that the set $\{\{a\},\{a,b\}\}$ corresponds to the ordered pair $(a,b)$. Indeed, you should prove that $\{\{a\},\{a,b\}\}=\{\{c\},\{c,d\}\}$ if and only if $a=c$ and $b=d$.

The induction is now easy to complete.