Advertisements
Advertisements
प्रश्न
Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = ab2 for all a, b ∈ Q ?
उत्तर
Commutativity :
\[\text{Let }a, b \in Q . \text{Then}, \]
\[a * b = a b^2 \]
\[b * a = b a^2 \]
\[\text{Therefore},\]
\[a * b \neq b * a\]
Thus, * is not commutative on Q.
Associativity :
\[\text{Let }a, b, c \in Q . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( b c^2 \right)\]
\[ = a \left( b c^2 \right)^2 \]
\[ = a b^2 c^4 \]
\[\left( a * b \right) * c = \left( a b^2 \right) * c\]
\[ = a b^2 c^2 \]
\[\text{Therefore},\]
\[a * \left( b * c \right) \neq \left( a * b \right) * c\]
Thus, * is not associative on Q.
APPEARS IN
संबंधित प्रश्न
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 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 Q, define a * b = ab + 1
For each binary operation * defined below, determine whether * is commutative or associative.
On Q, define a * b = `(ab)/2`
For each binary operation * defined below, determine whether * is commutative or associative.
On Z+, define a * b = ab
Is * defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M. of a and b a binary operation? Justify your answer.
Find which of the operations given above has identity.
Let A = N × N and * be the 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, if any.
State whether the following statements are true or false. Justify.
If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a
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 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 = ab
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.
The binary operation * : R × R → R is defined as a * b = 2a + b. Find (2 * 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 ?
Check the commutativity and associativity of the following binary operation '*' on Q defined by \[a * b = \frac{ab}{4}\] for all a, b ∈ Q ?
Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative ?
Find the identity element in the set of all rational numbers except −1 with respect to *defined by a * b = a + b + ab.
On the set Z of integers, if the binary operation * is defined by a * b = a + b + 2, then find the identity element.
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.
A binary operation * is defined on the set R of all real numbers by the rule \[a * b = \sqrt{ a^2 + b^2} \text{for all a, b } \in R .\]
Write the identity element for * on R.
Let * be a binary operation defined by a * b = 3a + 4b − 2. Find 4 * 5.
Let * be a binary operation on N given by a * b = HCF (a, b), a, b ∈ N. Write the value of 22 * 4.
Q+ is the set of all positive rational numbers with the binary operation * defined by \[a * b = \frac{ab}{2}\] for all a, b ∈ Q+. The inverse of an element a ∈ Q+ is ______________ .
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 __________ .
Which of the following is true ?
The number of binary operation that can be defined on a set of 2 elements is _________ .
The number of commutative binary operations that can be defined on a set of 2 elements is ____________ .
On Z, define * by (m * n) = mn + nm : ∀m, n ∈ Z Is * binary on Z?
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.
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?
Choose the correct alternative:
In the set Q define a ⨀ b = a + b + ab. For what value of y, 3 ⨀ (y ⨀ 5) = 7?
Is the binary operation * defined on Z (set of integer) by m * n = m – n + mn ∀ m, n ∈ Z commutative?
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 * be binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R. Then the operation * is ______.
The identity element for the binary operation * defined on Q ~ {0} as a * b = `"ab"/2` ∀ a, b ∈ Q ~ {0} is ______.
Which of the following is not a binary operation on the indicated set?
Subtraction and division are not binary operation on.
