Let * be a binary operation on Z defined by a * b = a + b – 4 for al

1.	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.
Answer» (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

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.

Answer»

(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