170 Representation of Propositions and Reasonings. 



merits ; when may wo infer from the four implications 

 cw?+ by : cd', &or + ay : c'cZ 3 cM y + dy r : ab f , and efa/ + cy' : «'£, 

 (1) that either X or Y has happened but not both, (2) that 

 both X and Y have happened or else neither of them ? What 

 combinations among the events A, B, C, D are impossible ? 



In other words, we are required to find in terms of a, b, c, d, 

 (1) the weakest antecedent of x'y + xy f 9 (2) the weakest ante- 

 cedent of xy + x'y', and (3) what combinations among the 

 statements a, b, c, d are inconsistent with the data. 



From the data 



(cuc + by : cd')(bx + ay : c r d)(cx' + chj r : ab r ){dx r + cy' \ a r b) 

 we get by mere inspection 



my : {a r V + ctf)(cfy + c'oT) : : a ! V*, 



x'y r : (c f d f + ab f )(c f d f + a f b) : : c'cV. 

 Hence 



xy + x'y' \ oJV + cfd'i 

 that is, 



(a + b)(c + d)\x r y + xy r (1) 



Again, 



x'y : (V + cd'){a! + c'cT){c' + ab')(d' + alb) 



: : (a'b' + a r cd'+b r c f d)(c r d' + a'bc' + ab'd r ) : : a'Vc'cV. 



From the symmetry of the conditions we thence get 



xy' : a'Vc'cV. 

 Hence 



x'y + xy' \ a!b'c'd'\ 

 that is, 



a + b + c + d: xy + x r y f . ....... (2) 



Also, since 



wy + afff + afy + arf=l 9 

 we get 



that is, omitting the redundant term underlined, 



(a + b)(c + d):0 (3) 



This completes the solution. From (1) we learn that we 

 may infer that one and one only of the events X and Y has 

 happened, provided we know that A or B has happened in con- 

 junction with C or D. But since (3) informs us that this 

 conjunction is impossible, it follows that the data are not snjfji- 



* The symbol : : expresses equivalence ; thus for a, : /3 : : y : : 6 read tl * 

 implies /3 ; /3 is equivalent to y, and y is equivalent to 8. 



