Advertisements
Advertisements
प्रश्न
Write the identity element for the binary operation * on the set R0 of all non-zero real numbers by the rule \[a * b = \frac{ab}{2}\] for all a, b ∈ R0.
उत्तर
Let e be the identity element in R0 with respect to * such that
\[a * e = a = e * a, \forall a \in R_0 \]
\[a * e = a \text{ and } e * a = a, \forall a \in R_0 \]
\[\text{Then} , \]
\[\frac{ae}{2} = a \text{ and }\frac{ea}{2} = a, \forall a \in R_0 \]
\[ae = 2a, \forall a \in R_0 \]
\[a\left( e - 2 \right) = 0, \forall a \in R_0 \]
\[e - 2 = 0, \forall a \in R_0 (\because a\neq0)\]
\[e = 2 \in R_0\]
Thus, 2 is the identity element in R0 with respect to *.
APPEARS IN
संबंधित प्रश्न
Show that the binary operation * on A = R – { – 1} defined as a*b = a + b + ab for all a, b ∈ A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible.
Determine whether or not of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.
On Z+, define * by a * b = ab
For each binary operation * defined below, determine whether * is commutative or associative.
On R − {−1}, define `a*b = a/(b+1)`
Let A = N × N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d)
Show that * is commutative and associative. Find the identity element for * on A, if any.
Consider a binary operation * on N defined as a * b = a3 + b3. Choose the correct answer.
(A) Is * both associative and commutative?
(B) Is * commutative but not associative?
(C) Is * associative but not commutative?
(D) Is * neither commutative nor associative?
Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by A * B = A ∩ B &mnForE; A, B in P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation*.
Determine whether the following operation define a binary operation on the given set or not : '⊙' on N defined by a ⊙ b= ab + ba for all a, b ∈ N
Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.
On R, define * by a * b = a + 4b2
Here, Z+ denotes the set of all non-negative integers.
Let * be a binary operation on the set I of integers, defined by a * b = 2a + b − 3. Find the value of 3 * 4.
Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.
Prove that the operation * on the set
\[M = \left\{ \begin{bmatrix}a & 0 \\ 0 & b\end{bmatrix}; a, b \in R - \left\{ 0 \right\} \right\}\] defined by A * B = AB is a binary operation.
Check the commutativity and associativity of the following binary operation '*'. on Z defined by a * b = a + b + ab for all a, b ∈ Z ?
Check the commutativity and associativity of the following binary operations '⊙' on Q defined by a ⊙ b = a2 + b2 for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on Z defined by a * b = a − b for all a, b ∈ Z ?
Check the commutativity and associativity of the following binary operation '*' on Q defined by \[a * b = \frac{ab}{4}\] for all a, b ∈ Q ?
If the binary operation * on the set Z is defined by a * b = a + b −5, the find the identity element with respect to *.
Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Show that '*' is both commutative and associative ?
Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Find the identity element in Z ?
Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element ?
Let R0 denote the set of all non-zero real numbers and let A = R0 × R0. If '*' is a binary operation on A defined by
(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A
Find the invertible element in A ?
Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}.
Define a binary operation on a set.
Let * be a binary operation on set of integers I, defined by a * b = 2a + b − 3. Find the value of 3 * 4.
If a * b = a2 + b2, then the value of (4 * 5) * 3 is _____________ .
Q+ denote the set of all positive rational numbers. If the binary operation a ⊙ on Q+ is defined as \[a \odot = \frac{ab}{2}\] ,then the inverse of 3 is __________ .
The binary operation * is defined by a * b = a2 + b2 + ab + 1, then (2 * 3) * 2 is equal to ______________ .
For the multiplication of matrices as a binary operation on the set of all matrices of the form \[\begin{bmatrix}a & b \\ - b & a\end{bmatrix}\] a, b ∈ R the inverse of \[\begin{bmatrix}2 & 3 \\ - 3 & 2\end{bmatrix}\] is ___________________ .
On the set Q+ of all positive rational numbers a binary operation * is defined by \[a * b = \frac{ab}{2} \text{ for all, a, b }\in Q^+\]. The inverse of 8 is _________ .
On Z, define * by (m * n) = mn + nm : ∀m, n ∈ Z Is * binary on Z?
Let A = `((1, 0, 1, 0),(0, 1, 0, 1),(1, 0, 0, 1))`, B = `((0, 1, 0, 1),(1, 0, 1, 0),(1, 0, 0, 1))`, C = `((1, 1, 0, 1),(0, 1, 1, 0),(1, 1, 1, 1))` be any three boolean matrices of the same type. Find (A ∧ B) v C
Is the binary operation * defined on Z (set of integer) by m * n = m – n + mn ∀ m, n ∈ Z commutative?
Let * be a binary operation defined on Q. Find which of the following binary operations are associative
a * b = ab2 for a, b ∈ Q
A binary operation on a set has always the identity element.
If the binary operation * is defined on the set Q + of all positive rational numbers by a * b = `" ab"/4. "Then" 3 "*" (1/5 "*" 1/2)` is equal to ____________.
Let * be a binary operation on set Q – {1} defind by a * b = a + b – ab : a, b ∈ Q – {1}. Then * is ____________.
Determine which of the following binary operation on the Set N are associate and commutaive both.
