Word Count : 726

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समुच्चय \( X = \mathbb{R} \setminus \{-1\} \) पर बाइनरी संक्रिया \( * \) को निम्नलिखित रूप में परिभाषित किया गया है: \[ X_1 * X_2 = X_1 + X_2 + X_1X_2, \quad \forall X_1, X_2 \in X \] अब, दिए गए प्रश्न के दोनों भागों को हल करते हैं। --- ### (i) जाँच: (X, *) एक समूह है या नहीं? किसी समुच्चय \( X \) को एक समूह होने के लिए निम्नलिखित गुणों को संतुष्ट करना आवश्यक है: 1. संयोजन गुण (Closure Property): हमें दिखाना होगा कि यदि \( X_1, X_2 \in X \) हैं, तो \( X_1 * X_2 \) भी \( X \) में होगा। चूँकि \( X_1, X_2 \neq -1 \), इसलिए: \[ X_1 * X_2 = X_1 + X_2 + X_1X_2 \] अब यदि \( X_1 * X_2 = -1 \), तो हमें देखना होगा कि क्या कोई \( X_1, _____ ____ __________ _________ ____ _________ _________ _________ _____ ______.
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