Sarthaks Test
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Let * be a binary operation on Z defined by a * b = a + b – 4 for all a, b ∈ Z.

(i) Show that * is both commutative and associative.

(ii) Find the identity element in Z

(iii) Find the invertible element in Z.

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(i) Let us prove the commutativity of *

Let a, b ∈ Z. Then,

a * b = a + b – 4

= b + a – 4

= b * a

So,

a * b = b * a, ∀ a, b ∈ Z

Thus, * is commutative on Z.

Now, let us prove associativity of Z.

Let a, b, c ∈ Z. Then,

a * (b * c) = a * (b + c – 4)

= a + b + c - 4 – 4

= a + b + c – 8

(a * b) * c = (a + b – 4) * c

= a + b – 4 + c – 4

= a + b + c – 8

So,

a * (b * c) = (a * b) * c, for all a, b, c ∈ Z

Thus, * is associative on Z.

(ii) Let e be the identity element in Z with respect to *

Such that, a * e = a = e * a ∀ a ∈ Z

a * e = a and e * a = a, ∀ a ∈ Z

a + e – 4 = a and e + a – 4 = a, ∀ a ∈ Z

e = 4, ∀ a ∈ Z

Thus, 4 is the identity element in Z with respect to *.

(iii) Let a and b ∈ Z be the inverse of a. Then,

a * b = e = b * a

a * b = e and b * a = e

a + b – 4 = 4 and b + a – 4 = 4

b = 8 – a ∈ Z

So, 8 – a is the inverse of a ∈ Z

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