Rule base — following IF-THEN with numbers
Use temperature and humidity membership degrees to calculate how strongly each IF-THEN rule fires with min / max.
The flow goes input membership → firing strength per rule → clipping the output peaks → aggregation → centroid.
Only 3 × 3 = 9 rules
In this course we have 3 temperature labels × 3 humidity labels, so the rule base has 9 rules. Laid out as a table:
| Temperature \ Humidity | Dry | Normal | Humid |
|---|---|---|---|
| Cold | Low | Low | Medium |
| Comfortable | Low | Medium | Medium |
| Hot | Medium | High | High |
The table by itself is close to plain-language rules. The fuzzy-control twist is that we use this table with membership degrees rather than 0/1.
Read AND as min
With inputs 26°C and 68%, Chapter 2 gave us the following membership degrees.
The firing strength of the rule "comfortable AND humid → medium" is min(0.50, 0.53) = 0.50. "Firing strength" is a value between 0 and 1 representing how strongly that rule applies; in some literature it is also called "degree of fulfillment" or "compatibility." This course uses "firing strength" as the standard translation. Taking AND as min keeps the intuition that "both sides have to be reasonably true before the rule fires strongly."
Note: using min for AND is one standard choice in the Mamdani method, and is one example of what fuzzy-logic literature calls a t-norm. Other t-norms such as the algebraic product (μA × μB) or the drastic product also exist, and different applications choose different ones. We use min throughout this course because it is easy to compute by hand, but please remember that "min is not the only choice."
Exercise 3-1 — Compute firing strength with min
Inputs: room temperature 26°C, humidity 68%. Use comfortable = 0.50, hot = 0.25, normal = 0.35, humid ≈ 0.53.
Q1. What is the firing strength of the rule 'comfortable AND humid → medium'?
AND = min, so min(0.50, 0.53) = 0.50.
Q2. What is the firing strength of the rule 'hot AND normal → high'?
AND = min, so min(0.25, 0.35) = 0.25.
Q3. For this input, how many rules have a firing strength greater than 0?
comfortable × normal, comfortable × humid, hot × normal, and hot × humid — all 4 rules fire at once.
Combine rules that map to the same output label with max
At 26°C and 68%, two rules map to medium and two rules map to high.
comfortable × normal → mediumis 0.35comfortable × humid → mediumis 0.50hot × normal → highis 0.25hot × humid → highis 0.25
On the output side, we combine medium-with-medium and high-with-high using max. That reduces them to two clipping heights: "medium 0.50, high 0.25." Max is the standard t-conorm implementation of OR, but other choices exist as well, such as the algebraic sum (μA + μB − μA·μB).
Exercise 3-2 — Aggregate same-output labels with max
Aggregate the firing strengths from the previous exercise by output label using max.
Q1. What is the aggregated height of the medium label?
The rules mapping to medium are 0.35 and 0.50, so taking max gives 0.50.
Q2. What is the aggregated height of the high label?
The rules mapping to high are 0.25 and 0.25, so aggregation still gives 0.25.