Advertisements
Advertisements
प्रश्न
Check the commutativity and associativity of the following binary operation '*'. on Z defined by a * b = a + b + ab for all a, b ∈ Z ?
Advertisements
उत्तर
Commutativity:
\[\text{Let a, b} \in Z . \text{Then}, \]
\[a * b = a + b + ab\]
\[ = b + a + ba\]
\[ = b * a \]
\[\text{Therefore},\]
\[a * b = b * a, \forall a, b \in Z\]
Associativity :
\[\text{Let } a, b, c \in Z . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( b + c + bc \right)\]
\[ = a + \left( b + c + bc \right) + a\left( b + c + bc \right)\]
\[ = a + b + c + bc + ab + ac + abc\]
\[\left( a * b \right) * c = \left( a + b + ab \right) * c\]
\[ = a + b + ab + c + \left( a + b + ab \right)c\]
\[ = a + b + ab + c + ac + bc + abc\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in Z\]
Thus, * is associative on Z.
APPEARS IN
संबंधित प्रश्न
Determine whether or not 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
For each binary operation * defined below, determine whether * is commutative or associative.
On Q, define a * b = ab + 1
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
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 the following operation define a binary operation on the given set or not : '⊙' on N defined by a ⊙ b= ab + ba 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.
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.
Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.
Let S = {a, b, c}. Find the total number of binary operations on S.
Find the total number of binary operations on {a, b}.
Let * be a binary operation on N given by a * b = LCM (a, b) for all a, b ∈ N. Find 5 * 7.
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 operation 'o' on Q defined by \[\text{a o b }= \frac{ab}{2}\] for all a, b ∈ Q ?
Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative ?
Let S be the set of all real numbers except −1 and let '*' be an operation defined by a * b = a + b + ab for all a, b ∈ S. Determine whether '*' is a binary operation on S. If yes, check its commutativity and associativity. Also, solve the equation (2 * x) * 3 = 7.
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 ?
Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Find the identity element in Z ?
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.
Write the identity element for the binary operation * on the set R0 of all non-zero real numbers by the rule \[a * b = \frac{ab}{2}\] for all a, b ∈ R0.
Write the total number of binary operations on a set consisting of two elements.
Let * be a binary operation defined by a * b = 3a + 4b − 2. Find 4 * 5.
If a binary operation * is defined on the set Z of integers as a * b = 3a − b, then the value of (2 * 3) * 4 is ___________ .
The number of commutative binary operations that can be defined on a set of 2 elements is ____________ .
Let A = ℝ × ℝ and let * be a binary operation on A defined by (a, b) * (c, d) = (ad + bc, bd) for all (a, b), (c, d) ∈ ℝ × ℝ.
(i) Show that * is commutative on A.
(ii) Show that * is associative on A.
(iii) Find the identity element of * in A.
Consider the binary operation * defined by the following tables on set S = {a, b, c, d}.
| * | a | b | c | d |
| a | a | b | c | d |
| b | b | a | d | c |
| c | c | d | a | b |
| d | d | c | b | a |
Show that the binary operation is commutative and associative. Write down the identities and list the inverse of elements.
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?
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.
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 + ab ∀ a, b ∈ Q
The identity element for the binary operation * defined on Q ~ {0} as a * b = `"ab"/2` ∀ a, b ∈ Q ~ {0} is ______.
Let * be a binary operation on Q, defined by a * b `= (3"ab")/5` is ____________.
Let A = N x N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Then * is ____________.
Let * be a binary operation on set Q – {1} defind by a * b = a + b – ab : a, b ∈ Q – {1}. Then * is ____________.
The identity element for the binary operation * defined on Q – {0} as a * b = `"ab"/2 AA "a, b" in "Q" - {0}` is ____________.
Let * be the binary operation on N given by a * b = HCF (a, b) where, a, b ∈ N. Find the value of 22 * 4.
