Comprehension check
Finally, walk one case end-to-end and verify that membership degree → firing strength → aggregation → output is a single continuous chain in your head. Edge cases (no rule fires, membership is 0) are also confirmed here.
The comprehensive review has 7 questions (2 of which check edge cases). If you have worked through the practice problems in Chapters 1 to 6, every calculation takes a shape you have already seen. When you get one wrong, the correct number and the reasoning appear right there.
Final exercise — Connect the whole flow
For the final exercise, take room temperature 22°C and humidity 40% and walk through membership → firing strength → aggregation → output together. Q5 uses the boundary preset for observation; Q6 and Q7 confirm edge cases (only one rule firing, no rule firing).
Q1. For the comfortable triangle tri(20,24,28), what is the membership degree at 22°C?
22°C sits on the left slope, so (22-20)/(24-20) = 0.50.
Q2. For the dry trapezoid trap(20,20,30,45), what is the membership degree at 40%?
40% sits on the right slope, so (45-40)/(45-30) = 0.33.
Q3. At 22°C and 40%, what is the aggregated height of the low label?
The maximum of the rules asserting low is 'comfortable × dry → low' at 0.33, so the aggregated height is 0.33.
Q4. With 22°C and 40%, what is the final output from the label-center approximation?
The approximation gives (20×0.33 + 50×0.25) / (0.33 + 0.25) ≈ 32.86%. The exact centroid method gives about 31.09%.
Q5. For the 'Watch the boundary' preset, approximately how many points does the fuzzy output exceed the threshold output by?
About 70.13% − 50% = 20.13 points. At the boundary, fuzzy control lifts the output smoothly starting just before the step.
Q6. Edge case: at room temperature 24°C and humidity 55% we have 'comfortable = 1.00, normal = 1.00' and every other label is 0. How many rules fire with strength greater than 0?
Because the apexes line up exactly, only 'comfortable × normal → medium' fires at 1.00. The other 8 rules each contain a membership of 0, so their min becomes 0. Stepping away from the boundary can narrow the active set down to a single rule.
Q7. Edge case: suppose for some input the aggregated heights become low = 0.00, medium = 0.00, high = 0.00. What is the denominator (area) of the centroid method?
If all clipping heights are 0, μ_agg(x) is 0 everywhere and the area Σμ is 0 too. In code, you guard against the divide-by-zero (e.g., area < 1e-9) and fall back to a safe value (such as 0%). It does not occur with the membership functions in this course, but the guard is essential when you adjust the labels.