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On (\mathbb{R}), (a*b=pa+qb). Which condition is necessary and sufficient for associativity?समुच्चय (\mathbb{R}) पर (a*b=pa+qb) है। यह संक्रिया साहचर्य होने के लिए कौन-सी शर्त पर्याप्त और आवश्यक है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: (p^2=p) and (q^2=q)(p^2=p) और (q^2=q)
Step 1: ((a*b)*c=p(pa+qb)+qc=p^2a+pqb+qc). Step 2: (a*(b*c)=pa+q(pb+qc)=pa+pqb+q^2c). Step 3: Equality for every (a,c) requires (p^2=p) and (q^2=q).चरण 1: ((a*b)*c=p(pa+qb)+qc=p^2a+pqb+qc)। चरण 2: (a*(b*c)=pa+q(pb+qc)=pa+pqb+q^2c)। चरण 3: हर (a,c) के लिए बराबरी हेतु (p^2=p) और (q^2=q) चाहिए।
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