| N(AB) | N(aB) | |
| ⎯⎯ | > | ⎯⎯ . |
| N(A) | N(a) |
This can be done by substituting
N(AB) + N(Ab) for N(A), &c.
Thus,
| N(AB) | N(aB) | ||
| ⎯⎯ | > | ⎯⎯ , | |
| N(A) | N(a) | ||
| N(a) | N(A) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ , |
| N(aB) | N(AB) | ||
| N(aB) + N(ab) | N(AB) + N(Ab) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ , |
| N(aB) | N(AB) | ||
| N(ab) | N(Ab) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ , |
| N(aB) | N(AB) | ||
| N(ab) | N(aB) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ , |
| N(Ab) | N(AB) | ||
| N(Ab) + N(ab) | N(AB) + N(aB) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ , |
| N(Ab) | N(AB) | ||
| N(b) | N(B) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ , |
| N(Ab) | N(AB) | ||
| N(AB) | N(Ab) | ||
| ∴ | ⎯⎯ | > | ⎯⎯ . |
| N(B) | N(b) |
356. Given the number (U) of objects in the Universe, and the number of objects in each of the classes x1, x2, x3, … xn, shew that the least number of objects in the class (x1x2x3 … xn)
= U − N (x1) − N (x2) − N (x3) … − N (xn). 410
where N (x1) means the number of things which are not x1; N (x2) means the number of things which are not x2; &c. [J.]
Given N (x1), N (x2), &c., the number of objects in the class (x1 or x2 … or xn) is greatest when no object belongs to any pair of the classes x1, x2, …; and in this case it = N (x1) + N (x2) … + N (xn).
Hence the least number in the contradictory class, x1x2x3 … xn,
= U − N (x1) − N (x2) … − N (xn).