Q.

The question asks to identify the invalid property regarding fuzzy relations and lambda-cuts.

The first three images correctly describe valid properties: nestedness ($R_\beta \subseteq R_\lambda$ for $\lambda \leq \beta$), union preservation ($(R \cup S)_\lambda = R_\lambda \cup S_\lambda$), and intersection preservation ($(R \cap S)_\lambda = R_\lambda \cap S_\lambda$).

The fourth image claims $(\tilde{R})_\lambda \neq R_\lambda$ except when $\lambda = 1$. This is incorrect because standard notation defines $(\tilde{R})_\lambda$ and $R_\lambda$ as the same crisp set derived from the fuzzy relation. Thus, this statement is not a valid property.