# Thread: Tuples defined as sets

1. ## Tuples defined as sets

Hi,

I am new here, so apologies for any miscommunication.

I am going through the book Discrete Structures, Logic, and Computability by James L. Hein

One of the problems defines tuples using sets:
1. ( ) = the empty set
2. (x) = {x}
3. (x,y) = {x,{x,y}}

It's the last item I do not understand - can someone explain it to me in an intuitive manner?

2. ## Here's my guess

Are you sure 3 shouldn't be (x,y)={{x},{x,y}} ?
Looks like they're defining an n-tuple as a set of sets. Maybe the pattern would be more obvious for a 5-tuple:

$(x_1,x_2,x_3,x_4,x_5)=
\{
\{x_1\},
\{x_1,x_2\},
\{x_1,x_2,x_3\},
\{x_1,x_2,x_3,x_4\},
\{x_1,x_2,x_3,x_4,x_5\}
\}$

3. Originally Posted by xibalba
I am going through the book Discrete Structures, Logic, and Computability by James L. Hein
One of the problems defines tuples using sets:
1. ( ) = the empty set
2. (x) = {x}
3. (x,y) = {x,{x,y}}

It's the last item I do not understand - can someone explain it to me in an intuitive manner?
About six weeks ago we had a similar question from a text by Hein.
I not did actually believe it could have been quoted correctly.
But I actually found an old pre-publication review copy of his 1996 Discrete Mathematics text.
Sure enough there was the notation.
I can tell that it not standard in mathematics. But Hein is writing for computer science majors.
That said, I will try to answer the question about tuples from a mathematical point of view.

An ordered pair $(a,b)$ differs from the set $\{a,b\}$ in that $\{a,b\}=\{b,a\}$ but $(a,b)\not= (b,a)$.
So we need to be able to define an ordered pair in terms of sets.
We do this way $(a,b)=\left\{\{a\},\{a,b\}\right\}$.
BTW, In my edition of Hein he use that basic idea and not what you wrote $\{a,\{a,b\}\}.$
So basically the singleton set tells us the first term of the pair and its ‘partner’ in the double set is the second term of the pair.

Now an ordered triple is defined as an ordered pair in which the first term is an ordered pair and the second is the third coordinate of the triple. Then the other pair tells the other two coordinates.

4. ## Yes!!!

Yes, you are correct - my bad -
(x,y) = {{x},{x,y}}

I am still trying to wrap my head around this though, so please correct me - In plain words it means that the tuple (x,y) is actually a set that contains a set with x and a set with x and y. Two questions:

1. Why wouldn't {{x,y}} work by itself, why {x} is necessary?
2. Tuples are supposed to be ordered - where in {{x},{x,y}} is the order part?

5. Originally Posted by xibalba
1. Why wouldn't {{x,y}} work by itself, why {x} is necessary?
2. Tuples are supposed to be ordered - where in {{x},{x,y}} is the order part?
Did you read my response?

6. Plato answered the first part of question 1 in his post:

Originally Posted by Plato
An ordered pair $(a,b)$ differs from the set $\{a,b\}$ in that $\textcolor{red}{ \{a,b\}=\{b,a\}}$ but $\textcolor{red}{(a,b)\not= (b,a)}$.
So we need to be able to define an ordered pair in terms of sets.
We do this way $(a,b)=\left\{\{a\},\{a,b\}\right\}$.
That quote also hints at the answer to number 2:
We define $(a,b)=\{\{a\},\{a,b\}\}$ and $(b,a)=\{\{b\},\{a,b\}\}$ because $\{\{a\},\{a,b\}\}\neq\{\{b\},\{a,b\}\}$ which is important because $(a,b) \neq (b,a)$.

as for the question why $\{x\}$ and not $x$, it's because $\{x,\{x,y\}\}$ a set of two different kinds of things-- $x$ is a number and $\{x,y\}$ is a set. It's more common to see a set written so that all the things inside are the same kind of thing: $\{\{x\},\{x,y\}\}$ is a set that contains two sets (it makes it much easier to prove certain things if you can assume that all the contents of a set are the same kind of thing).

7. OK, I think I am starting to get it. Please correct me if I am wrong:

something like {{a},{a,b},{a,b,c}} in words would mean A set that contains the set whose first element is 'a', and contains a set whose first two elements are 'a' and 'b', and contains a set whose first three elements are 'a', 'b', and 'c' - thereby imposing order?

Thanks.

8. Yes. The whole point of {{x}, {x,y}} is to say that there are two member is an ordered pair and that one of them is "distinguished"- that is that x is treated differently from y. That is the "order". Note that we could as easily have written the pair (x, y) as {{y}, {x,y}} with the second member "singled out"- which way we do it is a matter of convention. The point is that they are ordered- that being first in the pair is different from being second.