Please help me!

Problem 1: Recall from the lecture notes that the (Cartesian) product of two sets A, B, written A B, is the set {(a, b)| a ? A, b ? B}, i.e. the set of all ordered pairs with first entry in A and second in B. Determine which of the following are true and which are false; if they are true provide a proof, if false give a counterexample.

- If A B = B A implies A = B.

I did it 2 ways and both comments said it needed to be edited.

1st way:

If A X B = B XA , then A = B False Suppose A= 215, B= 21 3 . Then AXB = B X A = 2 (1, 1 )} and A = B V Suppose A= 213, B= 2 25. Then AXB= / ( 1, 2 )'s B x A = ( ( 2 , 1) '] but AXB + BX A which is false unless 1 =d If A = O and BJ any set, then A x B = B xA = , but A# B unless B is also of It's only true if both A, B are now-empty or both are emptyFix Problem 1( b) If AXB = BXA, then A= B. False Let A = 1 1 3 , B = 21, 23 Then A x B = { ( 1, 1), ( 1, 2 )'3 and B X A = { ( 1 , 1 ) , ( 21 173 Since ( 1, 2 ) # ( 21 1) , A X B # BXA So, if AXB = BXA, it does not imply A = B