Advertisements
Advertisements
प्रश्न
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 ?
उत्तर
Let e be the identity element. Then,
\[a * e = a = e * a, \forall a \in N\]
\[HCF\left( a, e \right) = a = HCF\left( e, a \right), \forall a \in N\]
\[ \Rightarrow HCF\left( a, e \right) = a, \forall a \in N\]
We cannot find e that satisfies this condition.
So, the identity element with respect to * does not exist in N.
APPEARS IN
संबंधित प्रश्न
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|
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
For each binary operation * defined below, determine whether * is commutative or associative.
On Z+, define a * b = 2ab
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 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+ define * by a * b = |a − b|
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.
Let S be the set of all rational numbers of the form \[\frac{m}{n}\] , where m ∈ Z and n = 1, 2, 3. Prove that * on S defined by a * b = ab is not a binary operation.
Determine which of the following binary operations are associative and which are commutative : * on Q defined by \[a * b = \frac{a + b}{2} \text{ for all a, b } \in Q\] ?
Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = ab2 for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = a + ab for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation'*' on Q defined by a * b = (a − b)2 for all a, b ∈ 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 a binary operation on S ?
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.
Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as \[a * b = \begin{cases}a + b & ,\text{ if a + b} < 6 \\ a + b - 6 & , \text{if a + b} \geq 6\end{cases}\]
Show that 0 is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 − a being the inverse of a.
Define a binary operation on a set.
Write the total number of binary operations on a set consisting of two elements.
Let * be a binary operation, on the set of all non-zero real numbers, given by \[a * b = \frac{ab}{5} \text { for all a, b } \in R - \left\{ 0 \right\}\]
Write the value of x given by 2 * (x * 5) = 10.
Write the composition table for the binary operation ×5 (multiplication modulo 5) on the set S = {0, 1, 2, 3, 4}.
If the binary operation * on the set Z of integers is defined by a * b = a + 3b2, find the value of 2 * 4.
If a * b = a2 + b2, then the value of (4 * 5) * 3 is _____________ .
If a * b denote the bigger among a and b and if a ⋅ b = (a * b) + 3, then 4.7 = __________ .
If the binary operation ⊙ is defined on the set Q+ of all positive rational numbers by \[a \odot b = \frac{ab}{4} . \text{ Then }, 3 \odot \left( \frac{1}{5} \odot \frac{1}{2} \right)\] is equal to __________ .
Subtraction of integers is ___________________ .
The law a + b = b + a is called _________________ .
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 number by *: a * b = `sqrt(a^2 + b^2)` find the identity element if exist in R with respect to *
Determine whether * is a binary operation on the sets-given below.
a * b = min (a, b) on A = {1, 2, 3, 4, 5}
Define an operation * on Q as follows: a * b = `(("a" + "b")/2)`; a, b ∈ Q. Examine the closure, commutative and associate properties satisfied by * on Q.
Fill in the following table so that the binary operation * on A = {a, b, c} is commutative.
| * | a | b | c |
| a | b | ||
| b | c | b | a |
| c | a | c |
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) ∧ C
Let A be Q\{1} Define * on A by x * y = x + y – xy. Is * binary on A? If so, examine the commutative and associative properties satisfied by * on A
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
Is the binary operation * defined on Z (set of integer) by m * n = m – n + mn ∀ m, n ∈ Z commutative?
The identity element for the binary operation * defined on Q ~ {0} as a * b = `"ab"/2` ∀ a, b ∈ Q ~ {0} is ______.
The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is ____________.
Subtraction and division are not binary operation on.
