मान लीजिए कि *, R पर  p, q ∈ R द्वारा परिभाषित एक द्विआधारी संक्रिया है। संक्रिया इस प्रकार है:

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AAI ATC Junior Executive 21 Feb 2023 Shift 2 Official Paper
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  1. क्रमविनिमेय लेकिन साहचर्य नहीं
  2. साहचर्य लेकिन क्रमविनिमेय नहीं
  3. क्रमविनिमेय और साहचर्य
  4. न तो साहचर्य और न ही क्रमविनिमेय

Answer (Detailed Solution Below)

Option 1 : क्रमविनिमेय लेकिन साहचर्य नहीं
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Detailed Solution

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दिया गया है:

मान लीजिए कि * R पर  p, q ∈ R द्वारा परिभाषित एक द्विआधारी संक्रिया है।

अवधारणा:

यदि  तो संबंध क्रमविनिमेय है।

यदि  तो संबंध साहचर्य है।

गणना:

मान लीजिए कि *, R पर  p, q ∈ R द्वारा परिभाषित एक द्विआधारी संक्रिया है।

क्रमविनिमेय:

मान लीजिए p, q ∈ R

अब,

अतः R क्रमविनिमेय है।

साहचर्य​:

मान लीजिए p, q, r ∈ R तब

फिर 

अतः R साहचर्य नहीं है।

अतः विकल्प (1) सही है।

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