Cartesian Product syntax in dictionary order relation definition

[Diffy]

Definition. Suppose that A and B are two sets with order relations $$<_A$$ and $$<_B$$ respectively. Define an order relation < on A x B by defining $$a_1 \ x \ b_1 < a_2 \ x \ b_2$$ if $$a_1 <_A a_2$$, or if $$a_1 = a_2$$ and $$b_1 <_B b_2$$. It is called the dictionary order relation on A X B.

OK. I think I am just confused by the syntax here. Up to this point Munkres has used (a , b) to denote an element of A x B. I think here he wants $$a_1 \ x \ b_1$$ to be what I am used to being $$(a_1 , b_1 )$$

I think possibly the reason he changed notation, is that in an order relation (a , b) = { x| a < x < b} and he doesn't want us to get confused.

I just need verification.

What do you think?

 

[D H]

Suppose set A is (1,2,3) while set B is (a,b,c). Members of A and B do not compare.

The reason this is called dictionary order is because this is essentially the way we sort words. Suppose you have two words. First you compare the first letter in each word to each other; game over if these letters differ. You only go on to the second letter if the first letters match. You go on to the third if the second letters match, and so on.

Now back to the original example. With this ordering, (1,b) < (1,c) < (2,a) (for example).

 

[tacman]

Yes, you are correct. The notation a_1 \ x \ b_1 is used to represent the element (a_1, b_1) in the Cartesian product A x B. This notation is commonly used to avoid confusion with the notation (a, b) which is used to represent an open interval in an order relation. So, in this case, (a_1, b_1) is just a different way of representing the same element in A x B. This is just a matter of notation and does not change the definition or the concept of the dictionary order relation. It is simply a way of denoting the elements in the Cartesian product while avoiding confusion with other notations.