Advertisements
Advertisements
प्रश्न
Check the commutativity and associativity of the following binary operations '*'. on Q defined by a * b = a − b for all a, b ∈ Q ?
Advertisements
उत्तर
Commutativity:
\[\text{Let } a, b \in Q . \text{Then}, \]
\[ a * b = a - b\]
\[b * a = b - a\]
\[\text{Therefore},\]
\[a * b \neq b * a\]
Thus, * is not commutative on Q.
Associativity :
\[\text{Let }a, b, c \in Q . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( b - c \right)\]
\[ = a - \left( b - c \right)\]
\[ = a - b + c\]
\[\left( a * b \right) * c = \left( a - b \right) * c\]
\[ = a - b - c\]
\[\text{Therefore},\]
\[a * \left( b * c \right) \neq \left( a * b \right) * c\]
Thus, * is not associative on Q.
APPEARS IN
संबंधित प्रश्न
Number of binary operations on the set {a, b} are
(A) 10
(B) 16
(C) 20
(D) 8
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+, 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 Z+, define * by a * b = a
Here, Z+ denotes the set of all non-negative integers.
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 = a + ab for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation'*' on Q defined by a * b = (a − b)2 for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on N, defined by a * b = ab for all a, b ∈ N ?
Check the commutativity and associativity of the following binary operation '*' on Z defined by a * b = a − b for all a, b ∈ Z ?
Check the commutativity and associativity of the following binary operation '*' on N defined by a * b = gcd(a, b) for all a, b ∈ N ?
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.
On Z, the set of all integers, a binary operation * is defined by a * b = a + 3b − 4. Prove that * is neither commutative nor associative on Z.
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 ?
Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.
Find the identity element in the set of all rational numbers except −1 with respect to *defined by a * b = a + b + ab.
If the binary operation * on the set Z is defined by a * b = a + b −5, the find the identity element with respect to *.
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 * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Find the identity element in Q − {−1} ?
Let 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by \[a o b = \frac{ab}{2}, \text{ for all a, b } \in Q_0\] :
Find the identity element in Q0.
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}.
Consider the binary operation 'o' defined by the following tables on set S = {a, b, c, d}.
| o | a | b | c | d |
| a | a | a | a | a |
| b | a | b | c | d |
| c | a | c | d | b |
| d | a | d | b | c |
Show that the binary operation is commutative and associative. Write down the identities and list the inverse of elements.
Define an associative binary operation on a set.
Write the inverse of 5 under multiplication modulo 11 on the set {1, 2, ... ,10}.
If a * b = a2 + b2, then the value of (4 * 5) * 3 is _____________ .
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 ______________ .
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 __________ .
Let * be a binary operation defined on set Q − {1} by the rule a * b = a + b − ab. Then, the identify element for * is ____________ .
An operation * is defined on the set Z of non-zero integers by \[a * b = \frac{a}{b}\] for all a, b ∈ Z. Then the property satisfied is _______________ .
A binary operation * on Z defined by a * b = 3a + b for all a, b ∈ Z, is ________________ .
Let * be a binary operation on Q+ defined by \[a * b = \frac{ab}{100} \text{ for all a, b } \in Q^+\] The inverse of 0.1 is _________________ .
The number of binary operation 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?
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 ∧ B
If the binary operation * is defined on the set Q + of all positive rational numbers by a * b = `" ab"/4. "Then" 3 "*" (1/5 "*" 1/2)` is equal to ____________.
The binary operation * defined on set R, given by a * b `= "a+b"/2` for all a, b ∈ R is ____________.
Let * be a binary operation on set Q – {1} defind by a * b = a + b – ab : a, b ∈ Q – {1}. Then * is ____________.
Subtraction and division are not binary operation on.
