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Home » Let * be a binary operation on Q – {-1} defined by a * b = a + b + ab for all a, b ∈ Q – {-1}. Then, (i) Show that * is both commutative and associative on Q – {-1} (ii) Find the identity element in Q – {-1}
Let * be a binary operation on Q – {-1} defined by a * b = a + b + ab for all a, b ∈ Q – {-1}. Then, (i) Show that * is both commutative and associative on Q – {-1} (ii) Find the identity element in Q – {-1}
Binary Operations | CBSE | Class 12 | Exercise 3.4 | Maths | RD Sharma
Let * be a binary operation on Q – {-1} defined by a * b = a + b + ab for all a, b ∈ Q – {-1}. Then, (i) Show that * is both commutative and associative on Q – {-1} (ii) Find the identity element in Q – {-1}

Answers:

(i)

Consider,

a, b ∈ Q – {-1}

a * b = a + b + ab

= b + a + ba

= b * a

a * b = b * a, ∀ a, b ∈ Q – {-1}

 

a * (b * c) = a * (b + c + b c)

= a + (b + c + b c) + a (b + c + b c)

= a + b + c + b c + a b + a c + a b c

(a * b) * c = (a + b + a b) * c

= a + b + a b + c + (a + b + a b) c

= a + b + a b + c + a c + b c + a b c

a * (b * c) = (a * b) * c, ∀ a, b, c ∈ Q – {-1}

Hence, * is associative on Q – {-1}.

(ii)

Consider,

e be the identity element in I+ with respect to *

a * e = a = e * a, ∀ a ∈ Q – {-1}

a * e = a and e * a = a, ∀ a ∈ Q – {-1}

a + e + ae = a and e + a + ea = a, ∀ a ∈ Q – {-1}

e + ae = 0 and e + ea = 0, ∀ a ∈ Q – {-1}

e (1 + a) = 0 and e (1 + a) = 0, ∀ a ∈ Q – {-1}

e = 0, ∀ a ∈ Q – {-1} [because a not equal to -1]

Hence, 0 is the identity element in Q – {-1} with respect to *.

i

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