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I have a question regarding set-builder notation.

I have a set {{0,1},{0,2},{0,3},...,{1,2},{1,3},{1,4},...,...}.

I wrote this in set builder notation as {{x,y}|x,yεℕ∧y>x}.

I've been told that its an incorrect form. If someone could explain to me why its wrong and help me get to the correct form I'd appreciate it.

Some useful information: I've been told elements of a set shouldn't have an order is the reason behind why it's wrong.

Asaf Karagila
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CraveLearning
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  • Okay so my answer is probably invalid in light of our comments on it so I'll delete it. You should probably edit your original post to include the reason why the professor said it was wrong (that he said that elements of a set shouldn't have an order), to make things clearer for future readers. My guess is that refers to the $y>x$, like it implying an order relation on the sets or something? Honestly I'm not sure what he means by that myself, and I'm struggling to think of a way to define the set without appealing to order. It's actually giving me a headache lol. – PrincessEev May 10 '19 at 01:08
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    Sorry about that, I added that to the main post. I appreciate you taking the time to look it over. – CraveLearning May 10 '19 at 01:11

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It is true that sets do not have order. If you change all the inner braces to parentheses, you will have a set of ordered pairs, and your specification $x \lt y$ will work fine. If you want a set of two element sets instead of a set of ordered pairs, you can change the $x \lt y$ to $x \neq y$. That will accept both $\{1,2\}$ and $\{2,1\}$, but they are the same thing, so you only get one of them in the outer set. It is probably worth some words to point that out.

Ross Millikan
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