Mark the correct alternative in the following question-For the binary operation - on - defined by a - b-a-b-1- the identity clement isa 0b -1c 1d 2

We have, a nbsp;* nbsp;b nbsp;= nbsp;a nbsp;+ nbsp;b nbsp;+ 1 Let e be the identity element of *. Then, a*e=a=e #160;*a #8658;a+e+1=a #86 ...

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Standard XII

Mathematics

Invertible Element Binary Operation

Mark the corr...

Question

Mark the correct alternative in the following question:

For the binary operation * on Z defined by a * b = a + b + 1, the identity clement is

(a) 0 (b) $-$ 1 (c) 1 (d) 2

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Solution

We have,

a * b = a + b + 1

Let e be the identity element of . Then,

$a e = a = e * a \Rightarrow a + e + 1 = a \Rightarrow e = a - a - 1 ∴ e = - 1$

Hence, the correct alternative is option (b).

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Similar questions

Q. For the binary operation * on Z defined by a * b = a + b + 1 the identity element is
(a) 0
(b) −1
(c) 1
(d) 2

Q. Consider the binary operation * defined on Q − {1} by the rule
a * b = a + b − ab for all a, b ∈ Q − {1}
The identity element in Q − {1} is
(a) 0
(b) 1
(c)

\frac{1}{2}

(d) −1

Q. On

Q - {1},

*

be the binary operation defined as

a * b = a b + 1.

Then the identity element for

*

Q - {1}

Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.

(i) On Z⁺, define * by a * b = a − b

(ii) On Z⁺, define * by a * b = ab

(iii) On R, define * by a * b = ab²

(iv) On Z⁺, define * by a * b = |a − b|

(v) On Z⁺, define * by a * b = a

For each binary operation * defined below, determine whether * is commutative or associative.

(i) On Z, define a * b = a − b

(ii) On Q, define a * b = ab + 1

(iii) On Q, define a * b

(iv) On Z⁺, define a * b = 2^ab

(v) On Z⁺, define a * b = a^b

(vi) On R − {−1}, define

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Mark the correct alternative in the following question: For the binary operation * on Z defined by a * b = a + b + 1, the identity clement is (a) 0 (b) -1 (c) 1 (d) 2

We have, a * b = a + b + 1 Let e be the identity element of *. Then, a*e=a=e *a⇒a+e+1=a⇒e=a-a-1∴ e=-1 Hence, the correct alternative is option (b).

Mark the correct alternative in the following question:

For the binary operation * on Z defined by a * b = a + b + 1, the identity clement is

(a) 0 (b) $-$ 1 (c) 1 (d) 2

We have,

a * b = a + b + 1

Let e be the identity element of . Then,

$a e = a = e * a \Rightarrow a + e + 1 = a \Rightarrow e = a - a - 1 ∴ e = - 1$

Hence, the correct alternative is option (b).