Page:An Investigation of the Laws of Thought (1854, Boole, investigationofl00boolrich).djvu/136

$\scriptstyle {x=0yz+{\frac {0}{0}}y{\bar {z}}+{\frac {c'^{2}}{c'^{2}}}{\bar {y}}z+{\frac {0}{0}}{\bar {y}}z}$^{[errata 1]};or,$\scriptstyle {x=(1-y)z+{\frac {0}{0}}y(1-z)+{\frac {0}{0}}(1-y)(1-z)}$;or,$\scriptstyle {x=(1-y)z+{\frac {0}{0}}(1-z)}$; the interpretation of which is, Wherever the property $\scriptstyle {A}$ is present, there either $\scriptstyle {C}$ is present and $\scriptstyle {B}$ absent, or $\scriptstyle {C}$ is absent. And inversely, Wherever the property $\scriptstyle {C}$ is present, and the property $\scriptstyle {B}$ absent, there the property $\scriptstyle {A}$ is present.

These results may be much more readily obtained by the method next to be explained. It is, however, satisfactory to possess different modes, serving for mutual verification, of arriving at the same conclusion.

4. We proceed to the second method.

Proposition II.

If any equations, $\scriptstyle {V_{1}=0}$, $\scriptstyle {V_{2}=0}$, &c., are such that the developments of their first members consist only of constituents with positive coefficients, those equations may be combined together into a single equivalent equation by addition.

For, as before, let $\scriptstyle {At}$ represent any term in the development of the function $\scriptstyle {V_{1}}$, $\scriptstyle {Bt}$ the corresponding term in the development of $\scriptstyle {V_{2}}$, and so on. Then will the corresponding term in the development of the equation (1)$\scriptstyle {V_{1}+V_{2}+}$&c.$\scriptstyle {=0}$, formed by the addition of the several given equations, be $\scriptstyle {(A+B+}$&c.$\scriptstyle {)t}$. But as by hypothesis the coefficients $\scriptstyle {A}$, $\scriptstyle {B}$, &c. are none of them negative, the aggregate coefficient $\scriptstyle {A+B}$, &c. in the derived equation will only vanish when the separate coefficients $\scriptstyle {A}$, $\scriptstyle {B}$, &c. vanish together. Hence the same constituents will appear in the development of the equation (1) as in the several equations $\scriptstyle {V_{1}=0}$, $\scriptstyle {V_{2}=0}$, &c. of the original system taken collectively, and therefore the interpretation of the equation (1) will be equiva-

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1. Correction: $\scriptstyle {x=0yz+{\frac {0}{0}}y{\bar {z}}+{\frac {c'^{2}}{c'^{2}}}{\bar {y}}z+{\frac {0}{0}}{\bar {y}}z}$ should be amended to $\scriptstyle {x=0yz+{\frac {0}{0}}y{\bar {z}}+{\frac {c'^{2}}{c'^{2}}}{\bar {y}}z+{\frac {0}{0}}{\bar {y}}{\bar {z}}}$: detail