Interpreting an Answer to Graph Structure Function

[slakedlime]

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 $\phi(x) = x_1 max (x_2, x_3, x_4) x_5$

2. The attempt at a solution
Two types of unbroken circuit are possible:

i) There is a connection from $x_1$ to $x_4$ (e.g. $x_1 → x_2 → x_4$ OR $x_1 → x_3 → x_2 → x_4$)
ii) The triangle $x_1,x_2, x_3$ is connected even if $x_4$ is broken. Hence, current can loop from $x_1 → x_2 → x_3$ or from $x_1 → x_3 → x2.$

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

The following part of the answer also confuses me:
$max (x_2, x_3, x_4)$

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

 

[lippyka]

Homework EquationsThis isn't a homework question. The attempt at a solutionTwo types of unbroken circuit are possible:i) There is a connection from x_1 to x_4 (e.g. x_1 → x_2 → x_4 OR x_1 → x_3 → x_2 → x_4)ii) The triangle x_1,x_2, x_3 is connected even if x_4 is broken. Hence, current can loop from x_1 → x_2 → x_3 or from x_1 → x_3 → x2.In the diagram, I can see four nodes x_1,x_2, x_3, x_4. Where does x_5 come from?The answer is likely that x_5 is assumed to be a constant, so that the output of the circuit is a function of the inputs x_1,x_2,x_3,x_4 and x_5. The following part of the answer also confuses me:max (x_2, x_3, x_4)This is likely because the function outputs the maximum value of x_2, x_3, and x_4. This makes sense as the circuit will only output a voltage if all the inputs are connected (i.e. x_2, x_3, and x_4 are all connected to x_1). Therefore, the output of the circuit will be the maximum value of the inputs (x_2, x_3, and x_4).