Question

दिये गए गए कथन को पढिए और सही विकल्प का चुनाव कीजिए: समुच्चय R – {1} पर एक द्विचर संक्रिया निम्न प्रकार परिभाषित है: a*b = a + b – ab, ∀ a, b ∈ R

1️⃣ (*) साहचर्य है
2️⃣ (*) क्रम-विनिमेय है
3️⃣ (*) क्रम-विनिमेय तथा साहचर्य दोनों है
4️⃣ उपरोक्त में से कोई नहीं​

Answer 1

SOLUTION

TO CHOOSE THE CORRECT OPTION

A binary operation on the set R – {1} is defined as follows: a*b = a + b – ab, ∀ a, b ∈ R

1. (*) is associative

2. (*) is commutative

3. (*) is both commutative and associative

4. None of the above

EVALUATION

Here it is given that a binary operation on the set R – {1} is defined as follows :

a*b = a + b - ab , ∀ a, b ∈ R

CHECKING FOR ASSOCIATIVE

Let a , b , c ∈ R

a * ( b * c)

= a * ( b + c - bc )

= a + ( b + c - bc ) - a( b + c - bc )

= a + b + c - ab - bc - ac + abc

Again

( a * b ) * c

= ( a + b - ab ) * c

= ( a + b - ab ) + c - c( a + b - ab )

= a + b + c - ab - bc - ac + abc

∴ a * ( b * c) = ( a * b ) * c

So * is associative

CHECKING FOR COMMUTATIVE

Let a , b ∈ R

Now

a*b = a + b - ab

b*a = b + a - ba = a + b - ab

So * is commutative

FINAL ANSWER

Hence the correct option is

3. (*) is both commutative and associative

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