Advertisements
Advertisements
प्रश्न
Determine which of the following binary operations are associative and which are commutative : * on Q defined by \[a * b = \frac{a + b}{2} \text{ for all a, b } \in Q\] ?
उत्तर
Commutativity :
\[\text{ Let } a, b \in N . \text{Then}, \]
\[a * b = \frac{a + b}{2}\]
\[ = \frac{b + a}{2}\]
\[ = b * a\]
\[\text{Therefore},\]
\[a * b = b * a, \forall a, b \in N\]
Thus, * is commutative on N.
Associativity:
\[\text{Let }a, b, c \in N . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( \frac{b + c}{2} \right)\]
\[ = \frac{a + \left( \frac{b + c}{2} \right)}{2}\]
\[ = \frac{2a + b + c}{4}\]
\[\left( a * b \right) * c = \left( \frac{a + b}{2} \right) * c\]
\[ = \frac{\left( \frac{a + b}{2} \right) + c}{2}\]
\[ = \frac{a + b + 2c}{4}\]
\[\text{Thus},a * \left( b * c \right) \neq \left( a * b \right) * c\]
\[\text{ If a} = 1, b = 2, c = 3\]
\[1 * \left( 2 * 3 \right) = 1 * \left( \frac{2 + 3}{2} \right)\]
\[ = 1 * \frac{5}{2}\]
\[ = \frac{1 + \frac{5}{2}}{2}\]
\[ = \frac{7}{4}\]
\[\left( 1 * 2 \right) * 3 = \left( \frac{1 + 2}{2} \right) * 3\]
\[ = \frac{3}{2} * 3\]
\[ = \frac{\frac{3}{2} + 3}{2}\]
\[ = \frac{9}{4}\]
\[\text { Therefore, }\exists \text{ a} = 1, b = 2, c = 3 \in \text{ N such that a} * \left( b * c \right) \neq \left( a * b \right) * c\]
Thus, * is not associative on N.
APPEARS IN
संबंधित प्रश्न
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|
State whether the following statements are true or false. Justify.
For an arbitrary binary operation * on a set N, a * a = ∀ a a * N.
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
Consider a binary operation * on N defined as a * b = 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; A, B in P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation*.
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 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.
Check the commutativity and associativity of the following binary operations '*'. on Q defined by a * b = a − b for all a, b ∈ Q ?
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 'o' on Q defined by \[\text{a o b }= \frac{ab}{2}\] for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation'*' on Q defined by a * b = ab + 1 for all a, b ∈ Q ?
Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative ?
The binary operation * is defined by \[a * b = \frac{ab}{7}\] on the set Q of all rational numbers. Show that * is associative.
Let * be a binary operation on Q0 (set of non-zero rational numbers) defined by \[a * b = \frac{ab}{5} \text{for all a, b} \in Q_0\]
Show that * is commutative as well as associative. Also, find its identity element if it exists.
Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that '*' is both commutative and associative on Q − {−1}.
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 +5 on set S = {0, 1, 2, 3, 4}.
Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}.
Write the multiplication table for the set of integers modulo 5.
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.
On the set Z of all integers a binary operation * is defined by a * b = a + b + 2 for all a, b ∈ Z. Write the inverse of 4.
For the binary operation multiplication modulo 5 (×5) defined on the set S = {1, 2, 3, 4}. Write the value of \[\left( 3 \times_5 4^{- 1} \right)^{- 1}.\]
If a * b denote the bigger among a and b and if a ⋅ b = (a * b) + 3, then 4.7 = __________ .
Q+ denote the set of all positive rational numbers. If the binary operation a ⊙ on Q+ is defined as \[a \odot = \frac{ab}{2}\] ,then the inverse of 3 is __________ .
Let * be a binary operation on R defined by a * b = ab + 1. Then, * is _________________ .
For the binary operation * defined on R − {1} by the rule a * b = a + b + ab for all a, b ∈ R − {1}, the inverse of a 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 _________ .
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 * .
Let * be an operation defined as *: R × R ⟶ R, a * b = 2a + b, a, b ∈ R. Check if * is a binary operation. If yes, find if it is associative too.
Determine whether * is a binary operation on the sets-given below.
(a * b) = `"a"sqrt("b")` is binary on R
Let A = {a + `sqrt(5)`b : a, b ∈ Z}. Check whether the usual multiplication is a binary operation on A
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?
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 existence of an identity, the existence of inverse properties for the operation * on A
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 a binary operation on the set of integers I, defined by a * b = a + b – 3, then find the value of 3 * 4.
a * b = `((a + b))/2` ∀a, b ∈ N is
