Defining unordered pairs in set theory

definitionelementary-set-theory

I am reading Naive Set Theory by Paul Halmos and am on section 3 (page 9) where he is talking about the axiom of pairing. In his explanation he states that a and b are two sets and A is the set containing a and b. He defines the unordered pair {a,b} as

$$ \{x \epsilon A: x=a \ \ or \ \ x=b\} $$

He then says that this set contains a and b.
I have three questions about this:

  1. How do we know that the set/unordered pair contains both of the sets a and b if the condition for the element x of the given set is that it is equal to a or equal to b (I'm interpreting the 'or' as the logical operator).

  2. If A had no elements other than the sets a and b, does A={a,b}?

  3. Is {a,b} read as 'the set containing a and b (and the empty set)'?

Best Answer

  1. You can verify $a$ satisfies the condition $x=a\lor x=b$, so $a\in A$. Similarly, $b\in A$.
  2. Yes.
  3. It has $a$ and $b$ as elements and nothing else, and in particular $\emptyset\not\in\{a,\,b\}$ (unless $a=\emptyset\lor b=\emptyset$).
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