Q. 15.0( 2 Votes )

Let * be a binary

Answer :

i. We are given with the set Z which is the set of integers.


A general binary operation is nothing but association of any pair of elements a, b from an arbitrary set X to another element of X. This gives rise to a general definition as follows:


A binary operation * on a set is a function * : A X A A. We denote * (a, b) as a * b.


Here the function *: Z X Z Z is given by a * b = a + b – 4
For the ‘ * ’ to be commutative, a * b = b * a must be true for all a, b belong to Z. Let’s check.


1. a * b = a + b – 4


2. b * a = b + a – 4


a * b = b * a (as shown by 1 and 2)


Hence ‘ * ’ is commutative.


For the ‘ * ’ to be associative, a * (b * c) = (a * b) * c must hold for every a, b, c Z.


3. a * (b * c) = a * (b + c – 4) = a + (b + c – 4) – 4 = a + b + c – 8


4. (a * b) * c = (a + b – 4) * c = (a + b – 4) + c – 4 = a + b + c – 8


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


Hence ‘ * ’ is associative.


ii. Identity Element: Given a binary operation *: Z X Z Z, an element e Z, if it exists, is called an identity of the operation * , if a * e = a = e * a a Z.


Let e be the identity element of Z.


Therefore, a * e = a
a + e – 4 = a
e – 4 = 0
e = 4


iii. Given a binary operation with the identity element e in A, an element a Z is said to be invertible with respect to the operation, if there exists an element b in Z such that a * b = e = b * a and b is called the inverse of a and is denoted by a–1.


Let us proceed with the solution.


Let b Z be the invertible elements in Z of a, here a Z.


a * b = e (We know the identity element from previous)
a + b – 4 = 4
b = 8 – a


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