By Karl Fink
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Extra resources for A Brief History Of Mathematics
X n ) be the equation expressing the value of the jth output variable as a function of e and those input variables from which there is a path to z not passing through e. Then a set of inputs (a f ,. . ,a n ) provides a Boolean derivative discrimination of e — e(x] < ,. . ,xi) from the hypothesis that e is stuck at 0 if and only if: And input values (aj. . a n ) provide a Boolean derivative discrimination of e = e(X]<... X]) from the hypothesis that e is stuck at 1 if and only if: In these equations, , for example, signifies the "Boolean h derivative" of the expression Zj(e, X j .
Xi) from the hypothesis that e is stuck at 0 if and only if: And input values (aj. . a n ) provide a Boolean derivative discrimination of e = e(X]<... X]) from the hypothesis that e is stuck at 1 if and only if: In these equations, , for example, signifies the "Boolean h derivative" of the expression Zj(e, X j . . , the function of x j . . xn that is equal to 1 if and only if a change in value of e changes the value of the expression. ON TESTING AND EVIDENCE 25 The basic idea behind the Boolean derivative method is the natural and correct one that a fault is detectable by an input if, were the circuit normal, the value of some output variable would be different for that input than it would be if the circuit contained the fault in question (but no other faults).
Indeed, the conjunction of (2), (3), and (4) is inconsistent with the hypothesis. (1) and (4) together entail that al is related to b1; for bj is the only element of B whose subscript is less than or equal to that of a^ But (2) and (3) together entail that aj is not related to bj. For by (2), a2 is related to bi. By the original description of A and B, &i^&2. And (3) asserts that no BOOTSTRAPPING WITHOUT BOOTSTRAPS 45 two distinct members of A are related to the same member of B; a2 has preempted bi, so bj is not available for aj.
A Brief History Of Mathematics by Karl Fink