Advertisements
Advertisements
प्रश्न
Let A = Q ✕ Q, where Q is the set of all rational numbers, and * be a binary operation defined on A by (a, b) * (c, d) = (ac, b + ad), for all (a, b) (c, d) ∈ A.
Find
(i) the identity element in A
(ii) the invertible element of A.
(iii)and hence write the inverse of elements (5, 3) and (1/2,4)
उत्तर
(i)
Let E=(x, y) be the identity element in A with respect to *,∀ x, y∈Q such that X*E=X=E*X,∀ X∈A
⇒X*E=X and E*X=X
⇒(ax, b+ay)=(a, b) and (xa, y+xb)=(a, b)
Considering (ax, b+ay)=(a, b)
⇒ax=a
⇒x=1 and b+ay=b
⇒y=0 [∵x=1]
Also,
Considering (xa, y+xb(a, b)
⇒xa=a
⇒x=1and y+xb=b
⇒y=0 [∵ x=1]∴ (1, 0) is the identity element in A with respect to *.
(ii)
Let F=(m, n) be the inverse in A,∀ m, n∈Q
X*F=E and F*X=E
⇒(am, b+an)=(1, 0) and (ma, n+mb)=(1, 0)
Considering (am, b+an)=(1, 0)
⇒am=1
⇒m=1/a and b+an=0
⇒n=−b/a
Also,
Considering (ma, n+mb)=(1, 0)
⇒ma=1
⇒m=1/a and n+mb=0
⇒n=−b/a (∵m=1/a)
∴The inverse of (a, b)∈A with respect to * is
(iii)Now let us find the inverse of elements
Hence , inverse of
And inverse of
संबंधित प्रश्न
Let * be a binary operation, on the set of all non-zero real numbers, given by
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 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 the following operation define a binary operation on the given set or not :
Let * be a binary operation on the set I of integers, defined by a * b = 2a + b − 3. Find the value of 3 * 4.
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 R defined by a * b = a + b − 7 for all a, b ∈ R ?
Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative ?
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 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 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 :
Find the invertible elements in A ?
Construct the composition table for ×5 on Z5 = {0, 1, 2, 3, 4}.
Define a binary operation on a set.
If the binary operation * on the set Z of integers is defined by a * b = a + 3b2, find the value of 2 * 4.
If G is the set of all matrices of the form
Let * be a binary operation defined on Q+ by the rule
The number of commutative binary operations that can be defined on a set of 2 elements is ____________ .
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?
Choose the correct alternative:
A binary operation on a set S is a function from
Choose the correct alternative:
Which one of the following is a binary operation on N?
Choose the correct alternative:
In the set R of real numbers ‘*’ is defined as follows. Which one of the following is not a binary operation on R?
Let * be a binary operation defined on Q. Find which of the following binary operations are associative
a * b =
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 binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R. Then the operation * is ______.
A binary operation on a set has always the identity element.
Let A = N x N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Then * is ____________.
