# Q1(c) Fuzzy set operations [5]

Subject: Intelligent Systems II

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/is2/#tee/papers/paper-reexam-2026/q1c

Canonical: https://deep-learning-study-room.vercel.app/read/is2/tee/papers/paper-reexam-2026--q1c/

Actual TEE re-exam · 100 marks · 3 hours · Q1 compulsory + any 4 of Q2–Q7 · every answer verified (key errors flagged)

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

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

**Final answer.** As in the last four rows. De Morgan check: ¬X ∪ ¬Y = 1 − (X ∩ Y) and ¬X ∩ ¬Y = 1 − (X ∪ Y) ✓.

## Calculate every element separately

### Element c1



$$
\neg X:1-0.7=0.3\qquad\neg Y:1-0.6=0.4
$$





$$
X\cup Y:\max(0.7,0.6)=0.7
$$





$$
X\cap Y:\min(0.7,0.6)=0.6
$$





$$
\neg X\cup\neg Y:\max(0.3,0.4)=0.4
$$





$$
\neg X\cap\neg Y:\min(0.3,0.4)=0.3
$$



### Element c2



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





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





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





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





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



### Element c3



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





$$
X\cup Y:\max(0.2,0.8)=0.8
$$





$$
X\cap Y:\min(0.2,0.8)=0.2
$$





$$
\neg X\cup\neg Y:\max(0.8,0.2)=0.8
$$





$$
\neg X\cap\neg Y:\min(0.8,0.2)=0.2
$$



### Element c4



$$
\neg X:1-0.4=0.6\qquad\neg Y:1-0.7=0.3
$$





$$
X\cup Y:\max(0.4,0.7)=0.7
$$





$$
X\cap Y:\min(0.4,0.7)=0.4
$$





$$
\neg X\cup\neg Y:\max(0.6,0.3)=0.6
$$





$$
\neg X\cap\neg Y:\min(0.6,0.3)=0.3
$$



### Element c5



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





$$
X\cup Y:\max(0.9,0.1)=0.9
$$





$$
X\cap Y:\min(0.9,0.1)=0.1
$$





$$
\neg X\cup\neg Y:\max(0.1,0.9)=0.9
$$





$$
\neg X\cap\neg Y:\min(0.1,0.9)=0.1
$$