Intelligent Systems II · M2 · mocks

Q1(a) · Fuzzy operations [2]

M2 mock 2 · 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 [0.1, 0.5, 1] and B has memberships [0.9, 0.4, 0]. Find A ∪ B, A ∩ B, Bᶜ and A ∖ B using standard max–min fuzzy operations. [2 marks · Practice]

Given

Elementu1u2u3
A0.10.51
B0.90.40

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.

\mu_{A\cup B}=\max(\mu_A,\mu_B)\qquad\mu_{A\cap B}=\min(\mu_A,\mu_B)LaTeX: \mu_{A\cup B}=\max(\mu_A,\mu_B)\qquad\mu_{A\cap B}=\min(\mu_A,\mu_B)
\mu_{B^c}=1-\mu_B\qquad\mu_{A\setminus B}=\min(\mu_A,1-\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
\mu_{B^c}(u_1)=1-0.9=0.1LaTeX: \mu_{B^c}(u_1)=1-0.9=0.1
\mu_{A\setminus B}(u_1)=\min(0.1,0.1)=0.1LaTeX: \mu_{A\setminus B}(u_1)=\min(0.1,0.1)=0.1
Step 3. Repeat for each element
ElementABUnionIntersectionBᶜA ∖ B
u10.10.9max(0.1, 0.9) = 0.9min(0.1, 0.9) = 0.11 − 0.9 = 0.1min(0.1, 0.1) = 0.1
u20.50.4max(0.5, 0.4) = 0.5min(0.5, 0.4) = 0.41 − 0.4 = 0.6min(0.5, 0.6) = 0.5
u310max(1, 0) = 1min(1, 0) = 01 − 0 = 1min(1, 1) = 1
Final answer. A ∪ B = [0.9, 0.5, 1.0]; A ∩ B = [0.1, 0.4, 0.0]; Bᶜ = [0.1, 0.6, 1.0]; A ∖ B = [0.1, 0.5, 1.0].