, which means thatx∈Bandy∈A. In summary, we deducethat ifxis any element ofAthenx∈Balso, and furthermore that ifyis anyelement ofBtheny∈A. But this is exactly our criteria for showing that twosets are equal, so we conclude that we must haveA=B. Clearly this conditionworks, for both sides ofA×B=B×Areduce to justA×A.By the way, taking the Cartesian product of a set with itself is a fairlycommon occurrence in mathematics; we have already seen one example abovewhen we wrote the plane asR×R. The setA×Awill also play an importantrole when we discuss relations and functions in a later chapter.a) List all the ordered pairs in a three by four table. The top rowwould contain (10,1) (10,2) (10,3) (10,4), and so on.b) Replace the left-hand side by including two extra Cartesian prod-ucts in the union: (A×C)∪(A×D)∪(B×C)∪(B×D).c) The ordered pairs are not identical; for instance, (1,3)∈A×Bbut (1,3)∈B×A.ThusA×B=B×Afor these sets.d) If there are no elements inB, then there is no way to create an ordered pair (x, y)withy∈B.The game is over as soon as all available letters and numbers appear at least oncesomewhere in the list of moves. WhenA={a, b, c}andB={1,2,3}, suppose the firstplayer writes down (a,1).If the second player matches neither of these characters,say by playing (b,3), then the first player should take the remaining two characters,which are (c,2) to win the game.However, if the second player does match one ofthe characters, say by playing (b,1), then the first player should continue to matchthat character by playing (c,1). The game must now last for exactly two more moves,causing the first player to win in this scenario as well.Analysis of the game withA={a, b, c, d}andB={1,2,3}is left to the reader.In general, it turns out that the second player has a winning strategy if at least oneof|A|and|B|is even, while the first player can always win if both|A|and|B|areodd. (Can you figure out the winning strategies?) Finally, try to find a nice way torepresent this game by putting markers on a rectangular grid whose rows are labeledwith the letters inAand whose columns are labeled by the numbers inB.

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42CHAPTER 2.SET THEORYExercises54. What can we deduce about setsAandBifA×B=∅?55. Write the definition of the Cartesian productA×Busing bar notation.56. Explain how a standard deck of cards illustrates a Cartesian product.