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Vol. XXIII, No. 3
105
This is equivalent to the cartesion product of (AxB) and C. (AXB) x C= [ (a1.b1.cl), (a2,b1,ci), (a1,b2,cl), (a2,62,cl),
(a1.b3.cl). (a2,63,cl), (al,b1,c2), (a2,61,c2),
(a1,b2,c2), (a1,63,c2), (a2,63,c2) ] (2) Development of Prastara in the leftword more :
It is evidently clear that First Prastara as well as Second Prastara works very beutifully to do the cartesion product of sets. It is also clear that not only (AxB) x C is ordered but its tupler are ordered also. These tupler are known as consinations in the combinatorial system given in GJK.
Thus we find the two devices to do the cartesion product of sets in GJK. But these devicer hold good to do the cartesion product of sets in the rightward more only, as we have seen. It is well known that the cartesion product of sets exists also in the leftword more.
Now our efforts are to illustrate that Prastara as a device holds good to do the cartesion product of sets in the leftward more. (2.1) Left First Prastara : Place respectively each one of the elements of B.
61 62 63 Cover each one of these with A.
A A A
61 62 63 or blal bla2 b2a1b2a2b3a1b3a2 Their is equivalent to the cartesion product of Bond A. BxA=[(bl.al), (b1,a2), (b2,al). (b2,a2), (13,al), (b3a2)]
Again place respectively each one of the elements of C and cover each of them with the set just obtained. (BxA)
(BxA)
C2 orciblal, clbla2, clb2al, clb2a2, clb3a1, clb3a2
c2b1al, c2b1a2, c2b2al, c2b2a2, c2b?al, c2b3a2 This is equivalent to the cartesion product of Cand (BxA) CX (BXA) = [(01,bl,al), (c1,61,22), (c1,b2,al), (01,b2,22),
(cl.b3,al), (c1,63,a2). (c2.b1,al). (c2,61,2),
(c2,62,al), (c2,62, a 2), (c2,63,al), (c2,63,a2)] (2.2) Left Second Prastrar: Place B as many times or the member of the elements is A. B
B Keep one by one the elements of A on the top of each B.
a 2
CI
al
B
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