CBSE 12th Standard Maths Subject Relations and Functions HOT Questions 6 Mark Questions With Solution 2021
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CBSE 12th Standard Maths Subject Relations and Functions HOT Questions 6 Mark Questions With Solution 2021
12th Standard CBSE
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Reg.No. :
Maths
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Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as \(a * b=\left\{\begin{array}{ll} a+b, & \text { if } a+b<6 \\ a+b-6 & \text { if } a+b \geq 6 \end{array}\right.\) show that 0 is the identity for this operation and each element of the set is invertible with 6 - a being the inverse of a.
(a)
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CBSE 12th Standard Maths Subject Relations and Functions HOT Questions 6 Mark Questions With Solution 2021 Answer Keys
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Let X = {0, 1, 2, 3, 4, 5}.
The operation * on X is defined as:
\(a * b=\left\{\begin{array}{ll} a+b, & \text { if } a+b<6 \\ a+b-6 & \text { if } a+b \geq 6 \end{array}\right.\)
An element e ∈ X is the identity element for the operation *, if a * e = a = e * a ∀ a ∈ X
For a ∈ X we observed that
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.
ie \(\left\{\begin{array}{ll} a+b=0=b+a & \text { if } a+b<6 \\ a+6-6=0=b+a-6 & \text { if } a+b \geq 6 \end{array}\right.\)
i.e.,
a = −b or b = 6 − a
But, X = {0, 1, 2, 3, 4, 5} and a, b ∈ X. Then, a ≠ −b.
