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On (\mathbb{R}), (a*b=2a+b). Does this operation have an identity element?समुच्चय (\mathbb{R}) पर (a*b=2a+b) है। क्या इस क्रिया में कोई तत्समक अवयव है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: Noनहीं
Step 1: A right identity would require (a*e=2a+e=a), giving (e=-a), dependent on (a). Step 2: A left identity would require (e*a=2e+a=a), giving (e=0). Step 3: Since there is no single two-sided identity, no identity exists.चरण 1: दायाँ तत्समक हो तो (a*e=2a+e=a), यानी (e=-a), जो (a) पर निर्भर है। चरण 2: बायाँ तत्समक हो तो (e*a=2e+a=a), इसलिए (e=0)। चरण 3: दोनों ओर एक ही स्थिर अवयव न मिले तो तत्समक नहीं होता।
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