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I was reading this excellent answer on this particular thread about ordered pair definition:

How can an ordered pair be expressed as a set?

At one point the accepted answer says:

The problem with your proposal is that it does not have the defining property we want for ordered pairs: for example, $\emptyset\neq\{\emptyset\}$, so we want $(\emptyset,\{\emptyset\}) \neq (\{\emptyset\},\emptyset)$. But in your proposal, we have: $$\begin{align*} (\emptyset,\{\emptyset\}) &= \Bigl\{ \{\emptyset\}, \bigl\{\{\emptyset\}\bigr\}, \bigl\{ \emptyset, \{\emptyset\}\bigr\}\Bigr\},\\ (\{\emptyset\},\emptyset) &= \Bigl\{ \bigl\{ \{\emptyset\}\bigr\}, \{\emptyset\}, \bigl\{ \{\emptyset\},\emptyset\bigr\}\Bigr\}; \end{align*}$$ so that $(\emptyset,\{\emptyset\}) = (\{\emptyset\},\emptyset)$.

I am not sure how the RHS of both the equations are equal here. What set operations are used to achieve this?

Community
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Luca
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    None, really: the two righthand sides have the same elements, so they’re the same set. The three elements are ${\varnothing}$, ${{\varnothing}}$, and ${\varnothing,{\varnothing}}={{\varnothing},\varnothing}$. The order of listing of elements is irrelevant. – Brian M. Scott Jan 05 '15 at 22:31
  • Thank you. I was still stuck on the ordered pair and got confused that the ordering in a set does not matter! – Luca Jan 05 '15 at 22:32

1 Answers1

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Remember that in a set, the order in which the elements are written is irrelevant. So in your two right hand sides,

  • the "first" element of RHS1 is identical with the "second" element of RHS2, namely, $\{\emptyset\}$;
  • the "second" element of RHS1 is identical with the "first" element of RHS2, namely, $\{\{\emptyset\}\}$;
  • the "last" elements in RHS1 and RHS2 are the same: they are written differently as $\{\emptyset,\{\emptyset\}\}$ and $\{\{\emptyset\},\emptyset\}$, but these are actually equal for similar reasons to the above two points.

So RHS1 and RHS2 are the same set.

The answer you linked is very good and detailed, but perhaps it is worthwhile making a simpler point. To specify an ordered pair, we just need to know two things: what are the two elements; and which one comes first.

David
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