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On (\mathbb{R}), (a*b=a+\mu b). Which values of (\mu) make this operation associative?समुच्चय (\mathbb{R}) पर (a*b=a+\mu b) है। यह संक्रिया साहचर्य होने के लिए (\mu) के कौन-से मान संभव हैं?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: (\mu=0) or (\mu=1)(\mu=0) या (\mu=1)
Step 1: ((a*b)*c=(a+\mu b)+\mu c=a+\mu b+\mu c). Step 2: (a*(b*c)=a+\mu(b+\mu c)=a+\mu b+\mu^2c). Equality needs (\mu c=\mu^2c) for every (c), so (\mu=\mu^2). Step 3: Thus (\mu=0) or (\mu=1).चरण 1: ((a*b)*c=(a+\mu b)+\mu c=a+\mu b+\mu c)। चरण 2: (a*(b*c)=a+\mu(b+\mu c)=a+\mu b+\mu^2c)। बराबरी के लिए (\mu c=\mu^2c) हर (c) पर चाहिए, इसलिए (\mu=\mu^2)। चरण 3: अतः (\mu=0) या (\mu=1)।
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