Question
Download Solution PDFमान लीजिए कि *, R पर \(\rm p * q=\frac{p+q}{2}, \forall \) p, q ∈ R द्वारा परिभाषित एक द्विआधारी संक्रिया है। संक्रिया इस प्रकार है:
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Detailed Solution
Download Solution PDFदिया गया है:
मान लीजिए कि * R पर \(\rm p * q=\frac{p+q}{2}, \forall \) p, q ∈ R द्वारा परिभाषित एक द्विआधारी संक्रिया है।
अवधारणा:
यदि \(\rm p*q = q*p\) तो संबंध क्रमविनिमेय है।
यदि \(\rm (p*q)*r = p*(q*r)\) तो संबंध साहचर्य है।
गणना:
मान लीजिए कि *, R पर \(\rm p * q=\frac{p+q}{2}, \forall \) p, q ∈ R द्वारा परिभाषित एक द्विआधारी संक्रिया है।
क्रमविनिमेय:
मान लीजिए p, q ∈ R
अब,
\(\rm p*q=\frac{p+q}{2}=\frac{q+p}{2}=q*p\)
अतः R क्रमविनिमेय है।
साहचर्य:
मान लीजिए p, q, r ∈ R तब
\(\rm (p*q)*r=\frac{p+q}{2}*r=\frac{p+q+2r}{4}\)
\(\rm p*(q*r)=p*\frac{q+r}{2}=\frac{2p+q+r}{4}\)
फिर \(\rm (p*q)*r\ne p*(q*r)\)
अतः R साहचर्य नहीं है।
अतः विकल्प (1) सही है।
Last updated on May 26, 2025
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