Intelligent Systems II · TEE · papers
Q7(b) Lambda-cut of a fuzzy relation [5]
Re-examination · 25 Jun 2024
Actual historical paper ·100 marks attempted /140 offered · Q1 compulsory + any four of Q2–Q7. Source defects and solution assumptions are explicitly identified.
For the fuzzy relation R below, find the λ-cut relations for λ=.003 and λ=.5.
LaTeX: R=\begin{bmatrix}1&0.1&0&0.4\\0.001&0.5&0.5&0.6\\0.3&1&0.5&0\\0&0.4&0.002&0.1\end{bmatrix}Answer
Given
A4×4 fuzzy relation with entries between 0 and 1.
To find
Two crisp 0/1 relations retaining entries at or above the threshold.
Formula and why it applies
A weak λ-cut includes an ordered pair when its membership is at least λ. Replace included entries by 1 and excluded entries by 0. A strong cut would use a strict inequality, but that is not requested.
LaTeX: (R_\lambda)_{ij}=\begin{cases}1&R_{ij}\geq\lambda\\0&R_{ij}<\lambda\end{cases}Solution
Step 1. Threshold at.003
Zero,.001 and.002 are below.003 and become 0. All other positive entries are at least.1, so become 1.
LaTeX: R_{.003}=\begin{bmatrix}1&1&0&1\\0&1&1&1\\1&1&1&0\\0&1&0&1\end{bmatrix}Step 2. Threshold at.5, including equality
Entries equal to.5 are retained. Values 1,.5,.6 become 1; values.4,.3,.1,.002,.001,0 become 0.
LaTeX: R_{.5}=\begin{bmatrix}1&0&0&0\\0&1&1&1\\0&1&1&0\\0&0&0&0\end{bmatrix}Step 3. Verify one included and one excluded entry
R₂₃=.5≥.5, so output(2,3)=1. R₄₃=.002<.003, so output(4,3)=0. The row/column labels and 4×4 shape do not change.