Intelligent Systems II · M2 · mocks
Q1(a) · Fuzzy operations [2]
M2 mock 3 · 10 marks
Q1(a)+Q1(b) compulsory; any two of Q2–Q4. Answers immediately below each question.
Confirmed M2 pattern: Q1(a) [2] and Q1(b) [2] are compulsory. Attempt any two of Q2, Q3 and Q4, each worth 3 marks. The paper offers 13 marks; you attempt 10. These are newly written practice questions, not claims about the exact upcoming paper. Solutions are directly below each question; the teaching working is deliberately longer than a concise exam answer.
Q1(a). On U = {u1, u2, u3}, A has memberships [0.7, 0.2, 0.9] and B has memberships [0.3, 0.8, 0.4]. Find A ∪ B, A ∩ B, Bᶜ and A ∖ B using standard max–min fuzzy operations. [2 marks · Practice]
Given
| Element | u1 | u2 | u3 |
|---|---|---|---|
| A | 0.7 | 0.2 | 0.9 |
| B | 0.3 | 0.8 | 0.4 |
To find
Four membership lists in exactly the same element order.
Formula and why it applies
Union is OR → max. Intersection is AND → min. Complement is NOT → 1 minus membership. Difference A ∖ B means A AND NOT B, so first complement B, then intersect with A.
LaTeX: \mu_{A\cup B}=\max(\mu_A,\mu_B)\qquad\mu_{A\cap B}=\min(\mu_A,\mu_B)
LaTeX: \mu_{B^c}=1-\mu_B\qquad\mu_{A\setminus B}=\min(\mu_A,1-\mu_B)Solution
Step 1. Align the same element
All operations compare the two memberships of the same u. Do not compare A at u₁ with B at u₂.
Step 2. Calculate one element completely
LaTeX: \mu_{B^c}(u_1)=1-0.3=0.7
LaTeX: \mu_{A\setminus B}(u_1)=\min(0.7,0.7)=0.7Step 3. Repeat for each element
| Element | A | B | Union | Intersection | Bᶜ | A ∖ B |
|---|---|---|---|---|---|---|
| u1 | 0.7 | 0.3 | max(0.7, 0.3) = 0.7 | min(0.7, 0.3) = 0.3 | 1 − 0.3 = 0.7 | min(0.7, 0.7) = 0.7 |
| u2 | 0.2 | 0.8 | max(0.2, 0.8) = 0.8 | min(0.2, 0.8) = 0.2 | 1 − 0.8 = 0.2 | min(0.2, 0.2) = 0.2 |
| u3 | 0.9 | 0.4 | max(0.9, 0.4) = 0.9 | min(0.9, 0.4) = 0.4 | 1 − 0.4 = 0.6 | min(0.9, 0.6) = 0.6 |