Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as
a + b = {a + b, if a + b < 6; a + b}
Show that zero is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 - a being the inverse of a

Solution:

Let X = {0, 1, 2, 3, 4, 5}

The operation * is defined as a + b

= {a + b, if a + b < 6; a + b}

An element e ∈ X is the identity element for the operation *,

if a * e = a = e * a ∀a ∈ X

For a ∈ X,

a * 0 = a + 0 = a [a ∈ X ⇒ a + 0 < 6]

0 * a = 0 + a = a [a ∈ X ⇒ 0 + a < 6]

⇒ a * 0 = a = 0 * a ∀a ∈ X

Thus, 0 is the identity element for the given operation *.

An element a ∈ X is invertible if there exists b ∈ X such that a * b = 0 = b * a.

{a + b = 0 = b + a, if a + b < 6; a + b - 6 = 0 = b + a - 6 if a + b ≥ 6}

⇒ a = - b or b = 6 - a

X = {0, 1, 2, 3, 4, 5} and a, b ∈ X.

Then a = 6 - b.

Therefore,

b = 6 - a is the inverse of a for all a ∈ X.

Inverse of an element a ∈ X, a -: 0 is 6 - a i.e.,

a - 1 = 6 - a

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise ME Question 14

Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as, a + b = {a + b, if a + b < 6; a + b}. Show that zero is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 - a being the inverse of a

Summary:

For the given binary operation * on the set {0, 1, 2, 3, 4, 5} as a + b = {a + b, if a + b < 6; a + b}, we have concluded that zero is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 - a being the inverse of a