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On (\mathbb{R}), (a*b=a+b-2ab). Which element has no inverse?समुच्चय (\mathbb{R}) पर (a*b=a+b-2ab) है। किस अवयव का प्रतिलोम अस्तित्व में नहीं है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: (\frac{1}{2})(\frac{1}{2})
Step 1: The identity is (0) because (a+0-0=a). Step 2: For inverse (x), (a+x-2ax=0), so (x(1-2a)=-a). Step 3: At (a=\frac{1}{2}), the coefficient becomes zero and the equation is impossible, so it has no inverse.चरण 1: तत्समक (0) है क्योंकि (a+0-0=a)। चरण 2: प्रतिलोम (x) के लिए (a+x-2ax=0), अतः (x(1-2a)=-a)। चरण 3: (a=\frac{1}{2}) पर गुणक शून्य हो जाता है और समीकरण असंभव बनता है, इसलिए इसका प्रतिलोम नहीं।
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