The Cardinality of a fuzzy set is:

Correct option is D

1. Definition (σ-count / fuzzy cardinality)
• Discrete finite $$U: |A|_\sigma=\sum_{x\in U}\mu_A(x) \in [0,n]$$​.
• Continuous/infinite $$U: |A|_\sigma=\int_U \mu_A(x)\,d\mu$$​ (depends on measure and support).
2. Examples
• If $$U=\{x_1,\dots,x_5\}$$​ and $$\mu_A=[1,0.4,0,0.6,0],$$​ then $$|A|_\sigma=1+0.4+0+0.6+0=2.0$$​ (finite, non-integer).
• If $$U=\mathbb{R}$$​ and $$\mu_A(x)=1$$​ on all $$\mathbb{R}$$​, the integral is infinite.
3. Applications: Estimating “how many” elements to a degree, e.g., fuzzy database queries (“about 3 items”), decision-making and summarization.
4. Advantages: Captures partial membership counts; more informative than crisp cardinality for vague categories.
5. Disadvantages / Caveats: Value depends on universe size, discretization and measure; may be non-integer and context-sensitive.

Additional Knowledge

• a) 0 — Wrong. Only true for the empty fuzzy set ($$\mu_A(x)=0$$​ for all x). The question doesn’t specify emptiness.
• b) finite — Wrong (not always). Finite holds for finite universes; but with infinite/continuous universes and nonzero support, the σ-count can be infinite.
• c) infinite — Wrong (not always). Infinite occurs only when the support/measure is unbounded (e.g., full real line with nonzero membership).