Advertisements
Advertisements
प्रश्न
On the set Z of integers, if the binary operation * is defined by a * b = a + b + 2, then find the identity element.
उत्तर
Let e be the identity element in Z with respect to * such that
\[a * e = a = e * a, \forall a \in Z\]
\[a * e = a \text{ and }e * a = a, \forall a \in Z\]
\[a + e + 2 = a \text{ and }e + a + 2 = a, \forall a \in Z\]
\[e = - 2 , \forall a \in Z\]
Thus, -2 is the identity element in Z with respect to *.
APPEARS IN
संबंधित प्रश्न
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 Q, define a * b = ab + 1
Find which of the operations given above has identity.
State whether the following statements are true or false. Justify.
For an arbitrary binary operation * on a set N, a * a = ∀ a a * N.
Number of binary operations on the set {a, b} are
(A) 10
(B) 16
(C) 20
(D) 8
Discuss the commutativity and associativity of binary operation '*' defined on A = Q − {1} by the rule a * b= a − b + ab for all, a, b ∊ A. Also find the identity element of * in A and hence find the invertible elements of A.
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 Z+, defined * by a * b = a − b
Here, Z+ denotes the set of all non-negative integers.
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 = ab2
Here, Z+ denotes the set of all non-negative integers.
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.
Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.
Let * be a binary operation on N given by a * b = LCM (a, b) for all a, b ∈ N. Find 5 * 7.
Let '*' be a binary operation on N defined by a * b = 1.c.m. (a, b) for all a, b ∈ N
Check the commutativity and associativity of '*' on N.
Check the commutativity and associativity of the following binary operation '*' on R defined by a * b = a + b − 7 for all a, b ∈ R ?
Check the commutativity and associativity of the following binary operation '*' on N, defined by a * b = ab for all a, b ∈ N ?
If the binary operation o is defined by aob = a + b − ab on the set Q − {−1} of all rational numbers other than 1, shown that o is commutative on Q − [1].
On Q, the set of all rational numbers a binary operation * is defined by \[a * b = \frac{a + b}{2}\] Show that * is not associative on Q.
Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b \[-\] ab, for all a, b \[\in\] S:
Prove that * is commutative as well as associative ?
Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Find the invertible elements in Z ?
On R − {1}, a binary operation * is defined by a * b = a + b − ab. Prove that * is commutative and associative. Find the identity element for * on R − {1}. Also, prove that every element of R − {1} is invertible.
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 ?
Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}.
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.
Define identity element for a binary operation defined on a set.
Write the composition table for the binary operation ×5 (multiplication modulo 5) on the set S = {0, 1, 2, 3, 4}.
If a * b = a2 + b2, then the value of (4 * 5) * 3 is _____________ .
An operation * is defined on the set Z of non-zero integers by \[a * b = \frac{a}{b}\] for all a, b ∈ Z. Then the property satisfied 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 _________ .
The number of binary operation that can be defined on a set of 2 elements is _________ .
Let '*' be a binary operation on N defined by
a * b = 1.c.m. (a, b) for all a, b ∈ N
Find 2 * 4, 3 * 5, 1 * 6.
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 * .
Determine whether * is a binary operation on the sets-given below.
(a * b) = `"a"sqrt("b")` is binary on R
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
Is the binary operation * defined on Z (set of integer) by m * n = m – n + mn ∀ m, n ∈ Z commutative?
Let R be the set of real numbers and * be the binary operation defined on R as a * b = a + b – ab ∀ a, b ∈ R. Then, the identity element with respect to the binary operation * is ______.
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.
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative
a * b = a – b ∀ a, b ∈ Q
Let A = N x N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Then * is ____________.
Let * be a binary operation on set Q – {1} defind by a * b = a + b – ab : a, b ∈ Q – {1}. Then * is ____________.
