Consider a fuzzy set old as defined below Old = {(20, 0.1), (30, 0.2), (40,…

Concept: The alpha-cut (α-level set) of a fuzzy set A on a universe X is the crisp subset of X containing every element whose membership degree meets or exceeds α: Aα = {x ∈ X : μ(x) ≥ α}. Because Aα is a crisp set, it can equivalently be written as a compact list of its members, or as an indicator (0/1) value attached to every element of the full universe X — both notations describe the same set.

Application: Apply α = 0.4 to every element of Old by comparing its given membership degree to 0.4:

1. 20 has membership 0.1, which is less than 0.4 — indicator value 0.

2. 30 has membership 0.2, which is less than 0.4 — indicator value 0.

3. 40 has membership 0.4, which equals 0.4 — indicator value 1.

4. 50 has membership 0.6, which is at least 0.4 — indicator value 1.

5. 60 has membership 0.8, which is at least 0.4 — indicator value 1.

6. 70 has membership 1, which is at least 0.4 — indicator value 1.

7. 80 has membership 1, which is at least 0.4 — indicator value 1.

Cross-check: Collecting only the elements with indicator 1 gives the compact set {40, 50, 60, 70, 80}. Writing every element of the universe together with its indicator value gives the equivalent pair form {(20, 0), (30, 0), (40, 1), (50, 1), (60, 1), (70, 1), (80, 1)} — the same crisp set, just displayed over the full universe instead of only the qualifying elements, which confirms the two notations agree.

The three remaining choices can each be checked against this definition:

• {(40, 0.4)}: keeps only one element and its original (non-indicator) membership degree, instead of every element whose membership is at least 0.4.

• {50, 60, 70, 80}: correctly identifies the elements with membership strictly greater than 0.4, but wrongly drops 40 — 40’s membership is exactly 0.4, so the inclusive definition μ(x) ≥ α requires it to stay in the cut.

• {(20, 0.1), (30, 0.2)}: lists the two elements with membership below 0.4 — the elements the α = 0.4 cut is meant to exclude, not include.

Result: The alpha-cut of Old at α = 0.4 is {40, 50, 60, 70, 80}, equivalently written as the full-universe indicator pairs {(20, 0), (30, 0), (40, 1), (50, 1), (60, 1), (70, 1), (80, 1)}.