Word Count : 726

---

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, __________ ___ ______ _______ _______ _________ _________ _______ ____ _______ ___.
_____ ___ _____ ___ _______ _________ ______ ________ ________ _________ ____ ______.
_____ ________ ___ ____ ________ __________ __________ ______ _________ ________ _______ ____.
_______ _____ ________ _______ __________ ________ ____.
___ __________ _______ _______ ____ ________ _______ ___.
_______ __________ ___ _________ _____ _____ ____.
_______ ______ _____ ___ _______ __________ _______ _________ ___ _________ _________ ___.
___ ________ ______ ____ __________ ______ _____ __________ ____ ________ ___ ____.
__________ _____ ______ __________ _____ _____.
_________ _________ __________ _______ ___ ____ ____ _________ _________ _____ _______ _____.
___ ___ ____ _______ _________ _______ _____ _________ ______.
__________ _____ ___ __________ _____ ____ _____ __________.
_______ __________ ___ _______ ____ ____ ____ _______ _________ _________ _______.
__________ _____ _______ _______ ______ _____ __________ _______ __________ _____.
______ ___ _____ ________ _________.
_________ __________ __________ ___ ______ ______ _____ _____ _______ _________ _________ _________.
________ ____ _____ _________ _______ _______ ______ ___.
_____ ___ ________ _____ _________ _________ ________ ____ __________ ___ ____ _________.
___ __________ __________ _________ __________ _____ ____ _________ _________ ____ _________ _______.
_____ ____ ____ ____ _________.
__________ ___ ______ ______ __________ _________ ____.
_____ ________ _____ ______ _____.
________ ________ __________ ________ __________ ____ _______.
______ __________ __________ _______ ____ _____ ___ ______ _______ _______.
____ ___ _____ _________ __________ ___ ___ _______ _____ _________ ___ _______.
___ _______ _____ ________ ______ ______ __________ __________.
__________ _______ _______ __________ ______ _________.
______ ___ ______ ___ __________ _______ ____ ______ _______ ____.
___ ___ _______ ______ _____ _______.
________ ___ _________ _________ _______ ______ ___ _________ ________ _____ ______ ________.
________ ________ ____ ___ ___ _______ ___ _______ _________ __________.
____ _____ __________ ____ _________.
___ _________ _____ ___ _______ __________.
________ ___ __________ _________ ____ _________ _______ _________.
_______ _________ _____ ________ ______ _____ _________.
____ _______ _________ __________ ________ ________ _______ _____ _____ ___ ______ ___.
_____ _______ ___ __________ __________ _______ _____ ___ _____ _______.
______ __________ _______ __________ ___ ______ __________ __________.
_______ __________ _____ _______ _________ ___ _______.
_____ __________ __________ _______ _____ ______ _______ _____ ______ ____.
________ _______ _______ _____ ______ _____.
___ ________ ___ ________ ________ _____ ___ ______ ________ ____ _______.
________ ________ ______ ____ ______ _____ ______ ____ ____ __________ ______ _____.
___ ____ ___ ______ _______ ___ __________ ___ ____ _____ ___ ____.
______ ___ _________ ________ _____ ___.
____ _________ ______ _____ ___ __________ _______ ______ ___ _____ _____ ___.
_____ ________ ___ __________ _______ ______.
________ ____ _______ _____ _______ ________ ________ ___ ____ ________ _____.
_________ _______ __________ ___ __________ ______ ___.
___ _______ ______ __________ _______ ______ __________ ________ ______ ___ ______ _____.
_________ _______ ____ ____ ______.
__________ _______ ___ _______ ___ _______ _______ ___ _______ ___ ______ _____.
_____ ____ _____ ________ ___ _____ ____ ________ ________ ______.
________ ___ ________ ___ _________ ____ ___ _________ _________.
______ ______ _____ _______ _________ ________ _____ _____ _______ _______ ______ __________.
________ __________ ____ ______ ______ ________.
_________ _________ _________ ______ ______ ______ ______ ________ ______ _____ _________.
___ ____ _____ _______ _____ ________ ____ ________ ____ ______.
____ _______ ______ __________ ___ ____ _____.
__________ _________ ________ __________ ______ ________ ______ _________.
________ ___ ______ __________ _________ ________.
____ _____ _____ _______ _________ ____ __________ _____ ________.
_________ ___ __________ _________ _____ ________ _____.
__________ _____ ________ __________ ________ ________.
______ ____ ___ _________ ______ ____.
_______ ___ _____ ______ _____.
Get Full Answer on WhatsApp