Intelligent Systems II · M2 · papers

Q3(b) Fuzzy relations and composition [10]

Re-examination · 31 Jan 2026

Longer past-paper practice restricted to Fuzzy, GA and PSO.

Question. A = {0.6/small + 0.4/medium + 0.1/large}, B = {0.4/slow + 0.8/moderate + 0.3/fast}, C = {0.5/small + 0.2/medium + 0.4/large}. Find R = A × B, S = C × B and the max–min composition S ∘ R. [10 marks]

Answer

Given

The fuzzy sets or relation matrices and their listed row/column order.

To find

The relation matrix/matrices and the composition specified in the question.

Formula and why it applies

For Cartesian products take min of row and column membership. For the ordered matrix composition shown here:

T_{ik}=\max_j\min(S_{ij},R_{jk})LaTeX: T_{ik}=\max_j\min(S_{ij},R_{jk})
If max–product is asked instead, replace the inner min by multiplication.

Solution

Step 1. R and S
R=\begin{bmatrix}0.4&0.6&0.3\\0.4&0.4&0.3\\0.1&0.1&0.1\end{bmatrix},\qquad S=\begin{bmatrix}0.4&0.5&0.3\\0.2&0.2&0.2\\0.4&0.4&0.3\end{bmatrix}LaTeX: R=\begin{bmatrix}0.4&0.6&0.3\\0.4&0.4&0.3\\0.1&0.1&0.1\end{bmatrix},\qquad S=\begin{bmatrix}0.4&0.5&0.3\\0.2&0.2&0.2\\0.4&0.4&0.3\end{bmatrix}
Step 2. S ∘ R
(1,1)=\max[\min(.4,.4),\min(.5,.4),\min(.3,.1)]=\max(.4,.4,.1)=0.4LaTeX: (1,1)=\max[\min(.4,.4),\min(.5,.4),\min(.3,.1)]=\max(.4,.4,.1)=0.4
(1,2)=\max[\min(.4,.6),\min(.5,.4),\min(.3,.1)]=\max(.4,.4,.1)=0.4LaTeX: (1,2)=\max[\min(.4,.6),\min(.5,.4),\min(.3,.1)]=\max(.4,.4,.1)=0.4
(1,3)=\max[\min(.4,.3),\min(.5,.3),\min(.3,.1)]=0.3LaTeX: (1,3)=\max[\min(.4,.3),\min(.5,.3),\min(.3,.1)]=0.3
\text{row 2 (all 0.2)}=(0.2,0.2,0.2);\quad(3,1)=\max(.4,.4,.1)=0.4,\;(3,2)=\max(.4,.4,.1)=0.4,\;(3,3)=0.3LaTeX: \text{row 2 (all 0.2)}=(0.2,0.2,0.2);\quad(3,1)=\max(.4,.4,.1)=0.4,\;(3,2)=\max(.4,.4,.1)=0.4,\;(3,3)=0.3
Final answer. R=\begin{bmatrix}.4&.6&.3\\.4&.4&.3\\.1&.1&.1\end{bmatrix}LaTeX: R=\begin{bmatrix}.4&.6&.3\\.4&.4&.3\\.1&.1&.1\end{bmatrix}, S=\begin{bmatrix}.4&.5&.3\\.2&.2&.2\\.4&.4&.3\end{bmatrix}LaTeX: S=\begin{bmatrix}.4&.5&.3\\.2&.2&.2\\.4&.4&.3\end{bmatrix}, S\circ R=\begin{bmatrix}0.4&0.4&0.3\\0.2&0.2&0.2\\0.4&0.4&0.3\end{bmatrix}LaTeX: S\circ R=\begin{bmatrix}0.4&0.4&0.3\\0.2&0.2&0.2\\0.4&0.4&0.3\end{bmatrix}. (The key gives R and S only.)

Every cell without the row shortcuts

Both printed relations run from size to speed; their middle labels do not match for genuine relation composition. The numerical working below follows the question’s intended row-of-S with column-of-R operation using explicitly assumed positional alignment of the three middle entries.

R: Cartesian-product cells

CellRow degreeColumn degreemin result
R110.60.40.4
R120.60.80.6
R130.60.30.3
R210.40.40.4
R220.40.80.4
R230.40.30.3
R310.10.40.1
R320.10.80.1
R330.10.30.1

S: Cartesian-product cells

CellRow degreeColumn degreemin result
S110.50.40.4
S120.50.80.5
S130.50.30.3
S210.20.40.2
S220.20.80.2
S230.20.30.2
S310.40.40.4
S320.40.80.4
S330.40.30.3

Result cell (1,1)

\max\{\min(0.4,0.4),\min(0.5,0.4),\min(0.3,0.1)\}LaTeX: \max\{\min(0.4,0.4),\min(0.5,0.4),\min(0.3,0.1)\}
=\max\{0.4,0.4,0.1\}=0.4LaTeX: =\max\{0.4,0.4,0.1\}=0.4

Result cell (1,2)

\max\{\min(0.4,0.6),\min(0.5,0.4),\min(0.3,0.1)\}LaTeX: \max\{\min(0.4,0.6),\min(0.5,0.4),\min(0.3,0.1)\}
=\max\{0.4,0.4,0.1\}=0.4LaTeX: =\max\{0.4,0.4,0.1\}=0.4

Result cell (1,3)

\max\{\min(0.4,0.3),\min(0.5,0.3),\min(0.3,0.1)\}LaTeX: \max\{\min(0.4,0.3),\min(0.5,0.3),\min(0.3,0.1)\}
=\max\{0.3,0.3,0.1\}=0.3LaTeX: =\max\{0.3,0.3,0.1\}=0.3

Result cell (2,1)

\max\{\min(0.2,0.4),\min(0.2,0.4),\min(0.2,0.1)\}LaTeX: \max\{\min(0.2,0.4),\min(0.2,0.4),\min(0.2,0.1)\}
=\max\{0.2,0.2,0.1\}=0.2LaTeX: =\max\{0.2,0.2,0.1\}=0.2

Result cell (2,2)

\max\{\min(0.2,0.6),\min(0.2,0.4),\min(0.2,0.1)\}LaTeX: \max\{\min(0.2,0.6),\min(0.2,0.4),\min(0.2,0.1)\}
=\max\{0.2,0.2,0.1\}=0.2LaTeX: =\max\{0.2,0.2,0.1\}=0.2

Result cell (2,3)

\max\{\min(0.2,0.3),\min(0.2,0.3),\min(0.2,0.1)\}LaTeX: \max\{\min(0.2,0.3),\min(0.2,0.3),\min(0.2,0.1)\}
=\max\{0.2,0.2,0.1\}=0.2LaTeX: =\max\{0.2,0.2,0.1\}=0.2

Result cell (3,1)

\max\{\min(0.4,0.4),\min(0.4,0.4),\min(0.3,0.1)\}LaTeX: \max\{\min(0.4,0.4),\min(0.4,0.4),\min(0.3,0.1)\}
=\max\{0.4,0.4,0.1\}=0.4LaTeX: =\max\{0.4,0.4,0.1\}=0.4

Result cell (3,2)

\max\{\min(0.4,0.6),\min(0.4,0.4),\min(0.3,0.1)\}LaTeX: \max\{\min(0.4,0.6),\min(0.4,0.4),\min(0.3,0.1)\}
=\max\{0.4,0.4,0.1\}=0.4LaTeX: =\max\{0.4,0.4,0.1\}=0.4

Result cell (3,3)

\max\{\min(0.4,0.3),\min(0.4,0.3),\min(0.3,0.1)\}LaTeX: \max\{\min(0.4,0.3),\min(0.4,0.3),\min(0.3,0.1)\}
=\max\{0.3,0.3,0.1\}=0.3LaTeX: =\max\{0.3,0.3,0.1\}=0.3