Solution

Answer: Option 3 is NOT valid.

Key Points 

Definitions & Notation

Let ~R and ~S be fuzzy relations on the same universe. Their membership functions are μ_R(x) and μ_S(x) in [0,1].

λ-cut (also called α-cut): For 0 ≤ λ ≤ 1, the λ-cut of a fuzzy relation ~R is the crisp relation

~R_λ = { x : μ_R(x) ≥ λ }.

Standard fuzzy set operations (Zadeh):

• Union: μ_{R ∪ S}(x) = max(μ_R(x), μ_S(x))
• Intersection: μ_{R ∩ S}(x) = min(μ_R(x), μ_S(x))

Check each option

Option 1:  (~R ∪ ~S)_λ = ~R_λ ∪ ~S_λ

Proof:

x ∈ (~R ∪ ~S)_λ ⇔ μ_R∪S(x) ≥ λ ⇔ max(μ_R(x),μ_S(x)) ≥ λ
⇔ (μ_R(x) ≥ λ) or (μ_S(x) ≥ λ) ⇔ x ∈ ~R_λ ∪ ~S_λ.

Hence, Option 1 is VALID.

Option 2:  (~R ∩ ~S)_λ = ~R_λ ∩ ~S_λ

Proof:

x ∈ (~R ∩ ~S)_λ ⇔ μ_R∩S(x) ≥ λ ⇔ min(μ_R(x),μ_S(x)) ≥ λ
⇔ (μ_R(x) ≥ λ) and (μ_S(x) ≥ λ) ⇔ x ∈ ~R_λ ∩ ~S_λ.

Hence, Option 2 is VALID.

Option 4:  For any λ ≤ β (0 ≤ β ≤ 1), it is true that R_β ⊆ R_λ

Reason: If μ_R(x) ≥ β and λ ≤ β, then automatically μ_R(x) ≥ λ. Thus every element that passes the stricter threshold β also passes the looser threshold λ.

Hence, Option 4 is VALID (monotonicity of λ-cuts).

Option 3:  (~R)_λ ≠ ~R_λ except when λ = 1

Why this is NOT valid: The notation (~R)_λ simply means “the λ-cut of the fuzzy relation ~R,” which by definition is exactly ~R_λ for every λ ∈ [0,1]. There is no special equality only at λ=1; the two notations denote the same crisp relation for all λ.

Conclusion: Options 1, 2, and 4 are standard λ-cut properties and are valid. Option 3 contradicts the definition of a λ-cut and is therefore the incorrect (not valid) statement.