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## Homework Statement

This isn't a homework question. I'm working through my book's exercises and am having difficulty interpreting an answer. Any guidance will be very much appreciated.

The problem is to come up with a structure function for a graph (image attached with this post). The answer is [itex]\phi(x) = x_1 max (x_2, x_3, x_4) x_5[/itex]

**2. The attempt at a solution**

Two types of unbroken circuit are possible:

i) There is a connection from [itex]x_1[/itex] to [itex]x_4[/itex] (e.g. [itex]x_1 → x_2 → x_4[/itex] OR [itex]x_1 → x_3 → x_2 → x_4[/itex])

ii) The triangle [itex]x_1,x_2, x_3[/itex] is connected even if [itex]x_4[/itex] is broken. Hence, current can loop from [itex]x_1 → x_2 → x_3[/itex] or from [itex]x_1 → x_3 → x2.[/itex]

In the diagram, I can see four nodes [itex]x_1,x_2, x_3, x_4[/itex]. Where does [itex]x_5[/itex] come from?

The following part of the answer also confuses me:

[itex]max (x_2, x_3, x_4)[/itex]

Why is this the case? I can't see how the circuit would function if [itex]x_2[/itex] and [itex]x_3[/itex] were both broken. Current simply could not flow from [itex]x_1[/itex] to [itex]x_4[/itex] unless component 2 or 3 were also working.