Intelligent Systems II · TEE · papers

Q1(c) Fuzzy set operations [5]

Final examination · 24 Nov 2025

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

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)LaTeX: \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
c1c2c3c4c5
X0.60.80.30.50.7
Y0.40.50.90.60.2
X ∪ Y = max0.60.80.90.60.7
X ∩ Y = min0.40.50.30.50.2
¬X ∪ ¬Y0.60.50.70.50.8
¬X ∩ ¬Y0.40.20.10.40.3
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=0.6LaTeX: \neg X:1-0.6=0.4\qquad\neg Y:1-0.4=0.6
X\cup Y:\max(0.6,0.4)=0.6LaTeX: X\cup Y:\max(0.6,0.4)=0.6
X\cap Y:\min(0.6,0.4)=0.4LaTeX: X\cap Y:\min(0.6,0.4)=0.4
\neg X\cup\neg Y:\max(0.4,0.6)=0.6LaTeX: \neg X\cup\neg Y:\max(0.4,0.6)=0.6
\neg X\cap\neg Y:\min(0.4,0.6)=0.4LaTeX: \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=0.5LaTeX: \neg X:1-0.8=0.2\qquad\neg Y:1-0.5=0.5
X\cup Y:\max(0.8,0.5)=0.8LaTeX: X\cup Y:\max(0.8,0.5)=0.8
X\cap Y:\min(0.8,0.5)=0.5LaTeX: X\cap Y:\min(0.8,0.5)=0.5
\neg X\cup\neg Y:\max(0.2,0.5)=0.5LaTeX: \neg X\cup\neg Y:\max(0.2,0.5)=0.5
\neg X\cap\neg Y:\min(0.2,0.5)=0.2LaTeX: \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=0.1LaTeX: \neg X:1-0.3=0.7\qquad\neg Y:1-0.9=0.1
X\cup Y:\max(0.3,0.9)=0.9LaTeX: X\cup Y:\max(0.3,0.9)=0.9
X\cap Y:\min(0.3,0.9)=0.3LaTeX: X\cap Y:\min(0.3,0.9)=0.3
\neg X\cup\neg Y:\max(0.7,0.1)=0.7LaTeX: \neg X\cup\neg Y:\max(0.7,0.1)=0.7
\neg X\cap\neg Y:\min(0.7,0.1)=0.1LaTeX: \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=0.4LaTeX: \neg X:1-0.5=0.5\qquad\neg Y:1-0.6=0.4
X\cup Y:\max(0.5,0.6)=0.6LaTeX: X\cup Y:\max(0.5,0.6)=0.6
X\cap Y:\min(0.5,0.6)=0.5LaTeX: X\cap Y:\min(0.5,0.6)=0.5
\neg X\cup\neg Y:\max(0.5,0.4)=0.5LaTeX: \neg X\cup\neg Y:\max(0.5,0.4)=0.5
\neg X\cap\neg Y:\min(0.5,0.4)=0.4LaTeX: \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=0.8LaTeX: \neg X:1-0.7=0.3\qquad\neg Y:1-0.2=0.8
X\cup Y:\max(0.7,0.2)=0.7LaTeX: X\cup Y:\max(0.7,0.2)=0.7
X\cap Y:\min(0.7,0.2)=0.2LaTeX: X\cap Y:\min(0.7,0.2)=0.2
\neg X\cup\neg Y:\max(0.3,0.8)=0.8LaTeX: \neg X\cup\neg Y:\max(0.3,0.8)=0.8
\neg X\cap\neg Y:\min(0.3,0.8)=0.3LaTeX: \neg X\cap\neg Y:\min(0.3,0.8)=0.3