हिंदी

Define Identity Element for a Binary Operation Defined on a Set. - Mathematics

Advertisements
Advertisements

प्रश्न

Define identity element for a binary operation defined on a set.

उत्तर

Let * be a binary operation on a set A.
An element e is called an identity element in A with respect to * if and only if

\[a * e = e * a = a, \forall a \in A\]

shaalaa.com
  क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
अध्याय 3: Binary Operations - Exercise 3.6 [पृष्ठ ३५]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12
अध्याय 3 Binary Operations
Exercise 3.6 | Q 10 | पृष्ठ ३५

संबंधित प्रश्न

Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table.

(i) Compute (2 * 3) * 4 and 2 * (3 * 4)

(ii) Is * commutative?

(iii) Compute (2 * 3) * (4 * 5).

(Hint: use the following table)

* 1 2 3 4 5
1 1 1 1 1 1
2 1 2 1 2 1
3 1 1 3 1 1
4 1 2 1 4 1
5 1 1 1 1 5

Consider a binary operation * on defined as 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; AB in P(X) is the power set of X. Show that is the identity element for this operation and is the only invertible element in P(X) with respect to the operation*.


If a * b denotes the larger of 'a' and 'b' and if a∘b = (a * b) + 3, then write the value of (5)∘(10), where * and ∘ are binary operations.


Determine whether the following operation define a binary operation on the given set or not :

\[' * ' \text{on Q defined by } a * b = \frac{a - 1}{b + 1} \text{for all a, b} \in Q .\]


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 Z+, define * by a * b = a

Here, Z+ denotes the set of all non-negative integers.


Let S = {abc}. Find the total number of binary operations on S.


Determine which of the following binary operation is associative and which is commutative : * on N defined by a * b = 1 for all a, b ∈ N ?


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 N defined by a * b = gcd(a, b) for all a, b ∈ N ?


On the set Q of all ration numbers if a binary operation * is defined by \[a * b = \frac{ab}{5}\] , prove that * is associative on Q.


Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.


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 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

Show that '*' is both commutative and associative on A ?


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 ?


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 +5 on set S = {0, 1, 2, 3, 4}.


Construct the composition table for ×5 on Z5 = {0, 1, 2, 3, 4}.


For the binary operation ×10 on set S = {1, 3, 7, 9}, find the inverse of 3.


Write the multiplication table for the set of integers modulo 5.


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.


On the power set P of a non-empty set A, we define an operation ∆ by

\[X ∆ Y = \left( \overline{X} \cap Y \right) \cup \left( X \cap \overline{Y} \right)\]

Then which are of the following statements is true about ∆.


If G is the set of all matrices of the form

\[\begin{bmatrix}x & x \\ x & x\end{bmatrix}, \text{where x } \in R - \left\{ 0 \right\}\] then the identity element with respect to the multiplication of matrices as binary operation, is ______________ .


Q+ is the set of all positive rational numbers with the binary operation * defined by \[a * b = \frac{ab}{2}\] for all ab ∈ Q+. The inverse of an element a ∈ Q+ is ______________ .


Which of the following is true ?


The binary operation * defined on N by a * b = a + b + ab for all a, b N is ________________ .


Let * be a binary operation on R defined by a * b = ab + 1. Then, * is _________________ .


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?


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 existence of identity and the existence of inverse for the operation * 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

Consider the binary operation * defined on the set A = {a, b, c, d} by the following table:

* a b c d
a a c b d
b d a b c
c c d a a
d d b a c

Is it commutative and associative?


In the set N of natural numbers, define the binary operation * by m * n = g.c.d (m, n), m, n ∈ N. Is the operation * commutative and associative?


Let * be a binary operation defined on Q. Find which of the following binary operations are associative

a * b = `"ab"/4` for a, b ∈ Q.


Consider the binary operation * on Q defind by a * b = a + 12b + ab for a, b ∈ Q. Find 2 * `1/3`.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×