By Charles L. Dodgson

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This set will be denoted by A x B and can be written as AxB = {(ayb): aeA and beB}. 4. Given two sets A and B, the cartesian product is the set of all OJ dered pairs (ayb) where aeA and beB. AxB It is not difficult to see that A x B # B x A unless A = 0 or B = 0 or y4 = B ; in the last case we may also write A2 in place of Ax A. T h e reader will have no difficulty in seeing that the cartesian plane is the cartesian product of the set of all real numbers with itself. C. Relations Usually a relation consists in associating objects of one kind with objects of another kind.

Thus this is a model in which the axiom of extension is not satisfied and must therefore be excluded from our theory. C. Subsets and the empty set From the two axioms of the last two sections, sets exist and each set is uniquely determined by its elements. We now consider sets whose elements are elements of another set. For example, elements of the set E of all even integers are elements of the set Z of all integers; in this case we say that E is a subset of the set Z. We formulate the general situation as a definition.

Given two relations R = (A,ByG) andS = (B,C,//), the composition of R and S is defined as the relation SoR = (A,Cf HoG)t where HoG = {(afc)eAxC: (a>b)eG and (byc)eH for some beB}. Notice that among the sets of departure and destination the set of departure of S is the set of destination of R, the set of departure of SoR is the set of departure of Rr and the set of destination of SoR is the set of destination of S. Furthermore, prxR z=> p^SoR, pr2S ZD pr2SoR. RELATIONS 48 [ Chap. 11. 9). Let G = {(3 cos 0 , 0 ) : 0e^Q, and H = {(2 sin 0, 0): OeX}, so that G is a cosine-curve and H i s a sine-curve.