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# Fault tree analysis & minimum cut set question

## Fault tree analysis & minimum cut set question

(OP)
I'm picking up FTA theory from "An Introduction to Reliability and Maintainability Enginenering", Charles Ebeling. In his final FTA example, he has an event consisting of the intersection of 3 events:

event = {wheel subsystem failure, b3, b4}

"Wheel subsystem failure" (wsf) is further expressible as:

wsf = (w1+b1)(w2+b2)(w3+b3)(w4+b4)

where I use algebraic multiplication and addition to represent set intersection and union, respectively. The b's represent 4 brake-pad assemblies (1 per wheel) and the w's represent 4 wheel cylinders. I re-expressed this as:

event = wsf b3 b4

= (w1+b1) (w2+b2) x
[ w3 w4 + b3 w4 + w3 b4 + b3 b4 ] b3 b4

The 2nd line can be simplified using the set theory identify (A+B+C)C = C, with C being b3.b4:

event = (w1+b1) (w2+b2) b3 b4

Since it decomposes into 4 terms in sum-of-products form, this should yield 4 minimum cut sets. But this disagrees with the statements in the textbook:

"The wheel subsystem failure can be decomposed into 16 combinations of wheel cylinder and brake-pad assembly failures. In the latter case, four of these decompositions include failure of both b3 and b4. Therefore, only 12 unique cut sets are formed."

I actually expanded out the wsf expression, and indeed I get 16 terms, 4 of which contain b3.b4. The only way that I know of to drop only those 4 terms is if b3.b4 was a cut set that was identified from another part of the tree. This is definitely not the case. So from my newbie perspective, my reduction of cut sets using (A+B+C)C=C is the only way, yielding far fewer minimum cut sets than implied by the author.

What am I missing?

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