So I was doing the exercise for section 3.5 and am having a hard time understanding a question.
Firstly, he defines an ordered pair as:
"If $x$ and $y$ are any objects (possibly equal), we define the ordered pair $(x, y)$ to be a new object, consisting of $x$ as its first component and $y$ as its second component. Two ordered pairs $(x, y)$ and $(x' , y')$ are considered equal if and only if both their components match, i.e.
$(x, y)=(x', y') \iff (x = x' \& \ y = y')$"
Now in the question he asks:
Suppose we define the ordered pair $(x, y)$ for any objects $x$ and $y$ by the formula $(x, y) := \{ \{x\}, \{x, y\}\}$. Show that such a definition indeed obeys the property of an ordered pair. For an additional challenge, show that the alternate definition $(x, y) := \{x, \{x, y\}\}$ also verifies the property and is thus also an acceptable definition of ordered pair.
My problem is understanding the difference in the question he is asking. What is the difference between showing the new definition obeys the property and verifying the property? Why will verifying the property help in allowing the new definition to be a acceptable one?
Any help answering my questions will be extremely appreciated! 🙂
Best Answer
There is no difference. He just does not want to use the same word. For both these definitions he wants you to prove the property of ordered pairs to show that both of these definitions are fine. Thus you can choose whatever formal definition you prefer (even though you will likely never use the formal definition again).