Paths and cuts of the bridge
The bridge has minimal path sets \{1,4\}, \{2,5\}, \{1,3,5\} and \{2,3,4\}. Write the structure function by the path formula, and expand it far enough to see the four path monomials. Say how the same system is drawn as a series arrangement of parallel modules, and as a parallel arrangement of series modules, if components may be duplicated in the drawing.
By the path formula,
\varphi(\mathbf x)=1-(1-x_1x_4)(1-x_2x_5)(1-x_1x_3x_5)(1-x_2x_3x_4).
Expanding, the terms of degree two and three are x_1x_4+x_2x_5+x_1x_3x_5+x_2x_3x_4, the four path monomials, followed by products of pairs of paths with alternating signs; reducing x_i^2=x_i gives the polynomial in the chapter.
The path formula draws the bridge as four series branches, 1–4, 2–5, 1–3–5 and 2–3–4, placed in parallel. The cut formula draws it as four parallel modules, \{1,2\}, \{4,5\}, \{1,3,5\} and \{2,3,4\}, placed in series. Both drawings repeat components, which is why the bridge itself is not a series–parallel system.