# Show that an ordered 4-tuple is also an ordered $m$-tuple for every positive integer $m$ less than 4.

I'm not really sure what this question is asking me to do. Can someone please explain this to me just a little bit more? Maybe give some examples.

This question was taken from Enderton's Elements of Set Theory.

• How is an ordered $m$-tuple defined in this book? – Mark Bennet Sep 16 '14 at 20:42
• @MarkBennet It's not really defined, but implied to be $\langle x_1,x_2,x_3,...,x_m\rangle$. – Skull-Face Sep 16 '14 at 20:47
• Well the definition is everything here, especially in a book on set theory, where an ordered tuple should really be a set. Are you sure that it is not defined by iterating the definition of an ordered pair? – Mark Bennet Sep 16 '14 at 20:53
• @MarkBennet Here is the quote from the book were it starts to talk about n-tuples: "We can extend the ideas behind ordered pairs to the case of ordered triples and, more generally, to ordered $n$-tuples." It then defines triples as $\langle x,y,z\rangle = \langle\langle x,y\rangle,z\rangle$ and quadruples as $\langle x_1,x_2,x_3,x_4\rangle = \langle\langle x_1,x_2,x_3\rangle,x_4\rangle=\langle\langle\langle x_1,x_2\rangle,x_3\rangle,x_4\rangle$. – Skull-Face Sep 16 '14 at 20:59

Here is a hint:

Suppose we have the ordered $3$-tuple $\langle x_1, x_2, x_3\rangle=\langle\langle x_1, x_2\rangle, x_3\rangle$.

What happens if we define $y_1=\langle x_1, x_2\rangle$ and $y_2=x_3$?

• Then $\langle y_1,y_2\rangle=\langle\langle x_1,x_2\rangle,x_3\rangle=\langle x_1,x_2,x_3\rangle$, right? – Skull-Face Sep 16 '14 at 21:13
• Thanks. I think you gave me enough guidance for me to do the problem. – Skull-Face Sep 16 '14 at 21:14
• @Skull-Face I'm glad you got it. – Mark Bennet Sep 16 '14 at 21:15

A complete solution is:Take,$y_4=\langle x_1, x_2, x_3, x_4\rangle$ , $y_3=\langle x_1, x_2, x_3\rangle$ and $y_2=\langle x_1, x_2\rangle$ . Now, $\langle x_1, x_2, x_3, x_4\rangle=\langle\langle x_1, x_2, x_3\rangle, x_4\rangle=\langle y_3, x_4\rangle$, which is a two tuple. Again, $\langle x_1, x_2, x_3, x_4\rangle=\langle\langle x_1, x_2, x_3\rangle, x_4\rangle=\langle\langle \langle x_1, x_2\rangle, x_3\rangle, x_4\rangle=\langle\langle y_2, x_3\rangle, x_4\rangle=\langle y_2, x_3, x_4\rangle$ (by Definition), which is a triplet. By definition, $\langle y_4\rangle=y_4=\langle x_1, x_2, x_3, x_4\rangle$. Hence $\langle x_1, x_2, x_3, x_4\rangle=\langle y_4 \rangle$, which is a one tuple. Is this ans correct?