Advertisements
Advertisements
Question
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.
Solution
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
RELATED QUESTIONS
LetA= R × R and * be a 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. Also find the inverse of every element (a, b) ε A.
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
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.
On Z+, define * by a * b = |a − b|
For each binary operation * defined below, determine whether * is commutative or associative.
On Z, define a * b = a − b
For each binary operation * defined below, determine whether * is commutative or associative.
On R − {−1}, define `a*b = a/(b+1)`
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?
Define a binary operation *on the set {0, 1, 2, 3, 4, 5} as
a * b = `{(a+b, "if a+b < 6"), (a + b - 6, if a +b >= 6):}`
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.
Determine whether the following operation define a binary operation on the given set or not : '*' on N defined by a * b = a + b - 2 for all a, b ∈ N
Determine whether the following operation define a binary operation on the given set or not : '×6' on S = {1, 2, 3, 4, 5} defined by
a ×6 b = Remainder when ab is divided by 6.
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.
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 'o' on Q defined by \[\text{a o b }= \frac{ab}{2}\] for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on Q defined by \[a * b = \frac{ab}{4}\] for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on N defined by a * b = gcd(a, b) for all a, b ∈ N ?
On Q, the set of all rational numbers, * is defined by \[a * b = \frac{a - b}{2}\] , shown that * is no associative ?
On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b − 4. Prove that * is neither commutative nor associative on Z.
The binary operation * is defined by \[a * b = \frac{ab}{7}\] on the set Q of all rational numbers. Show that * is associative.
Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that '*' is both commutative and associative on Q − {−1}.
Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows (a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 × R :
Show that '⊙' is commutative and associative on A ?
Let * be the binary operation on N defined by a * b = HCF of a and b.
Does there exist identity for this binary operation one N ?
If a * b denote the bigger among a and b and if a ⋅ b = (a * b) + 3, then 4.7 = __________ .
Which of the following is true ?
The binary operation * is defined by a * b = a2 + b2 + ab + 1, then (2 * 3) * 2 is equal to ______________ .
A binary operation * on Z defined by a * b = 3a + b for all a, b ∈ Z, is ________________ .
Let * be a binary operation on Q+ defined by \[a * b = \frac{ab}{100} \text{ for all a, b } \in Q^+\] The inverse of 0.1 is _________________ .
Let * be a binary operation defined on Q+ by the rule
\[a * b = \frac{ab}{3} \text{ for all a, b } \in Q^+\] The inverse of 4 * 6 is ___________ .
If * is defined on the set R of all real numbers by *: a*b = `sqrt(a^2 + b^2 ) `, find the identity elements, if it exists in R with respect to * .
Examine whether the operation *defined on R by a * b = ab + 1 is (i) a binary or not. (ii) if a binary operation, is it associative or not?
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 v B
Let A be Q\{1}. Define * on A by x * y = x + y – xy. Is * binary on A? If so, examine the existence of an identity, the existence of inverse properties for the operation * on A
Choose the correct alternative:
A binary operation on a set S is a function from
Choose the correct alternative:
If a * b = `sqrt("a"^2 + "b"^2)` on the real numbers then * is
Is the binary operation * defined on Z (set of integer) by m * n = m – n + mn ∀ m, n ∈ Z commutative?
Let N be the set of natural numbers. Then, the binary operation * in N defined as a * b = a + b, ∀ a, b ∈ N has identity element.
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 ____________.
The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is ____________.
Let * be a binary operation on the set of integers I, defined by a * b = a + b – 3, then find the value of 3 * 4.
If * is a binary operation on the set of integers I defined by a * b = 3a + 4b - 2, then find the value of 4 * 5.
a * b = `((a + b))/2` ∀a, b ∈ N is
