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This paper solves the issues of determining the number <i>F_n</i> of fuzzy subsets of a nonempty finite set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>X</mi></semantics></math></inline-formula>. To solve this, this paper incorporates the equivalence relation on the collection of all fuzzy subsets of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>X</mi></semantics></math></inline-formula>. We derive two closed explicit formulas for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>, which is the sum of a finite series in the product of binomial numbers or the sum of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>k</mi></semantics></math></inline-formula>-level fuzzy subsets <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></mrow></semantics></math></inline-formula> by introducing a classification technique. Moreover, these explicit formulas enable us to find the number of the maximal chains of crisp subsets of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>.</mo></mrow></semantics></math></inline-formula> Further, this paper presents some elementary properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>F</mi><mi>n</mi></msub></mrow></semantics></math></inline-formula>.