Expert: Sherry Wallin - 6/18/2010

Question
Hi sir,
   given A=(2,3,5,7),B=(1,12,13,15) How many elements are there in A cross B and B cross A.Is A croos B equal to B cross A.Find the subsets of A cross 6 according to the relation "is less than"?

Iam having doubt if A croos B and B cross A is equal and how to find the subsets of A cross 6 according to relation "x less than y"

Answer
Hi Ann~
    I am assuming you want to find the Cartesian Cross Product and if so, then you are looking for the number of ordered pairs created by A x B. This is just the multiplication counting principle: if there are m things in the first set and n things in the 2nd set then there are mn things in the resulting set, in other words there are 4(4) = 16 ordered pairs that can be made. You can see this when 2 is the first element in the ordered pair there are 4 choices for the 2nd element and the same is true for each of the first elements being paired with each of the 2nd elements. Clearly A x B and B x A are the same in size but not equal because we are talking about ordered pairs and in B x A the first element in the ordered pairs comes from B and the 2nd element comes from A which is clearly different. As for the last question, (I think there is a typo and what you meant to say is find the subsets of A cross B (not 6) hopefully because that is how I am going to answer the question) you want to look at all the 16 ordered pairs that you get with
A x B and choose only the ones that have the first element less than the 2nd element of the ordered pair.
An easy way to do this is to look at the 2 in A and how many numbers in B are bigger than 2? I see 3. Take the 2nd element in A, that is 3 and ask how many elements in B are bigger than 3 and I see 3 more. Next look at the 5 in set A and see how many are bigger than 5 and again I see 3 and finally looking at the 7 in set A I see 3 that are bigger than 7. So there are 3+3+3+3 = 12 ordered pairs that satisfy the relation 'are less than' in A cross B. Note here each ordered pair is a subset. If instead you wrote the problem correctly and you really want A cross 6 then (2,6), (3,6), (5,6) are the only ordered pairs that satisfy the relation.

Feel free to ask more questions if this isn't clear.

Math Prof