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On (A={1,2,3,4,5}), how many equivalence relations have exactly two equivalence classes?समुच्चय (A={1,2,3,4,5}) पर ऐसे समतुल्यता संबंधों की संख्या कितनी है जिनमें ठीक दो समतुल्यता वर्ग हों?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: (15)(15)
Step 1: Exactly two classes means splitting (5) elements into two non-empty parts. Step 2: There are (2^5-2=30) non-empty proper subsets, but each partition is counted twice. Step 3: Hence the number is (\frac{30}{2}=15).चरण 1: ठीक दो वर्ग बनाने का अर्थ है (5) अवयवों को दो अरिक्त भागों में बांटना। चरण 2: अरिक्त उचित उपसमुच्चय (2^5-2=30) हैं, पर हर विभाजन दो बार गिना जाता है। चरण 3: इसलिए संख्या (\frac{30}{2}=15) है।
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