# Q7(b) Lambda-cut of a fuzzy relation [5]

Subject: Intelligent Systems II

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/is2/#tee/papers/paper-2024-reexam/q7b

Canonical: https://deep-learning-study-room.vercel.app/read/is2/tee/papers/paper-2024-reexam--q7b/

Actual historical paper ·100 marks attempted /140 offered · Q1 compulsory + any four of Q2–Q7. Source defects and solution assumptions are explicitly identified.

**Question. Q7(b)** [5 marks · 2024-06-25]

For the fuzzy relation R below, find the λ-cut relations for λ=.003 and λ=.5.



$$
R=\begin{bmatrix}1&0.1&0&0.4\\0.001&0.5&0.5&0.6\\0.3&1&0.5&0\\0&0.4&0.002&0.1\end{bmatrix}
$$



### 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.



$$
(R_\lambda)_{ij}=\begin{cases}1&R_{ij}\geq\lambda\\0&R_{ij}<\lambda\end{cases}
$$



#### 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.



$$
R_{.003}=\begin{bmatrix}1&1&0&1\\0&1&1&1\\1&1&1&0\\0&1&0&1\end{bmatrix}
$$



##### 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.



$$
R_{.5}=\begin{bmatrix}1&0&0&0\\0&1&1&1\\0&1&1&0\\0&0&0&0\end{bmatrix}
$$



##### 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.

**Final answer.** The two crisp matrices above are the weak λ-cuts.