# Q1(c) Fuzzy set operations [5]

Subject: Intelligent Systems II

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/is2/#tee/mocks/paper-final-2025/q1c

Canonical: https://deep-learning-study-room.vercel.app/read/is2/tee/mocks/paper-final-2025--q1c/

100-mark timed practice using the actual Final 2025 paper. Q1 compulsory; any four of Q2–Q7. Worked solutions are below the questions.

**Question. Consider two fuzzy sets X = {(c1, 0.6), (c2, 0.8), (c3, 0.3), (c4, 0.5), (c5, 0.7)} and Y = {(c1, 0.4), (c2, 0.5), (c3, 0.9), (c4, 0.6), (c5, 0.2)}. Find 1) X ∪ Y 2) X ∩ Y 3) ¬X ∪ ¬Y 4) ¬X ∩ ¬Y.** [5 marks]

### Answer

#### Given

The aligned X and Y memberships printed in this question.

#### To find

The four requested membership lists in the original element order.

#### Formula and why it applies

Use max for OR, min for AND, and 1−μ for NOT.



$$
\mu_{X\cup Y}=\max(\mu_X,\mu_Y),\quad\mu_{X\cap Y}=\min(\mu_X,\mu_Y)
$$



Complement both operands before applying the operator in ¬X∪¬Y or ¬X∩¬Y.

#### Solution

##### Step 1. Complements

¬X = {0.4, 0.2, 0.7, 0.5, 0.3}; ¬Y = {0.6, 0.5, 0.1, 0.4, 0.8}.

##### Step 2. Table

|  | c1 | c2 | c3 | c4 | c5 |
| --- | --- | --- | --- | --- | --- |
| X | 0.6 | 0.8 | 0.3 | 0.5 | 0.7 |
| Y | 0.4 | 0.5 | 0.9 | 0.6 | 0.2 |
| **X ∪ Y = max** | 0.6 | 0.8 | 0.9 | 0.6 | 0.7 |
| **X ∩ Y = min** | 0.4 | 0.5 | 0.3 | 0.5 | 0.2 |
| **¬X ∪ ¬Y** | 0.6 | 0.5 | 0.7 | 0.5 | 0.8 |
| **¬X ∩ ¬Y** | 0.4 | 0.2 | 0.1 | 0.4 | 0.3 |

##### Step 3. Check with De Morgan's law

¬X ∪ ¬Y must equal 1 − (X ∩ Y) = {0.6, 0.5, 0.7, 0.5, 0.8} ✓, and ¬X ∩ ¬Y must equal 1 − (X ∪ Y) = {0.4, 0.2, 0.1, 0.4, 0.3} ✓.

**Final answer.** X ∪ Y = {(c1,0.6),(c2,0.8),(c3,0.9),(c4,0.6),(c5,0.7)}; X ∩ Y = {(c1,0.4),(c2,0.5),(c3,0.3),(c4,0.5),(c5,0.2)}; ¬X ∪ ¬Y = {(c1,0.6),(c2,0.5),(c3,0.7),(c4,0.5),(c5,0.8)}; ¬X ∩ ¬Y = {(c1,0.4),(c2,0.2),(c3,0.1),(c4,0.4),(c5,0.3)}.

Official marking: 1 + 1 + 1.5 + 1.5.

## Calculate every element separately

### Element c1



$$
\neg X:1-0.6=0.4\qquad\neg Y:1-0.4=0.6
$$





$$
X\cup Y:\max(0.6,0.4)=0.6
$$





$$
X\cap Y:\min(0.6,0.4)=0.4
$$





$$
\neg X\cup\neg Y:\max(0.4,0.6)=0.6
$$





$$
\neg X\cap\neg Y:\min(0.4,0.6)=0.4
$$



### Element c2



$$
\neg X:1-0.8=0.2\qquad\neg Y:1-0.5=0.5
$$





$$
X\cup Y:\max(0.8,0.5)=0.8
$$





$$
X\cap Y:\min(0.8,0.5)=0.5
$$





$$
\neg X\cup\neg Y:\max(0.2,0.5)=0.5
$$





$$
\neg X\cap\neg Y:\min(0.2,0.5)=0.2
$$



### Element c3



$$
\neg X:1-0.3=0.7\qquad\neg Y:1-0.9=0.1
$$





$$
X\cup Y:\max(0.3,0.9)=0.9
$$





$$
X\cap Y:\min(0.3,0.9)=0.3
$$





$$
\neg X\cup\neg Y:\max(0.7,0.1)=0.7
$$





$$
\neg X\cap\neg Y:\min(0.7,0.1)=0.1
$$



### Element c4



$$
\neg X:1-0.5=0.5\qquad\neg Y:1-0.6=0.4
$$





$$
X\cup Y:\max(0.5,0.6)=0.6
$$





$$
X\cap Y:\min(0.5,0.6)=0.5
$$





$$
\neg X\cup\neg Y:\max(0.5,0.4)=0.5
$$





$$
\neg X\cap\neg Y:\min(0.5,0.4)=0.4
$$



### Element c5



$$
\neg X:1-0.7=0.3\qquad\neg Y:1-0.2=0.8
$$





$$
X\cup Y:\max(0.7,0.2)=0.7
$$





$$
X\cap Y:\min(0.7,0.2)=0.2
$$





$$
\neg X\cup\neg Y:\max(0.3,0.8)=0.8
$$





$$
\neg X\cap\neg Y:\min(0.3,0.8)=0.3
$$