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
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
Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table.
(i) Compute (2 * 3) * 4 and 2 * (3 * 4)
(ii) Is * commutative?
(iii) Compute (2 * 3) * (4 * 5).
(Hint: use the following table)
| * | 1 | 2 | 3 | 4 | 5 |
| 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 2 | 1 | 2 | 1 |
| 3 | 1 | 1 | 3 | 1 | 1 |
| 4 | 1 | 2 | 1 | 4 | 1 |
| 5 | 1 | 1 | 1 | 1 | 5 |
State whether the following statements are true or false. Justify.
For an arbitrary binary operation * on a set N, a * a = ∀ a a * N.
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 for all a, b ∈ N.
Determine whether the following operation define a binary operation on the given set or not :
\[' * ' \text{on Q defined by } a * b = \frac{a - 1}{b + 1} \text{for all a, b} \in Q .\]
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.
Prove that the operation * on the set
\[M = \left\{ \begin{bmatrix}a & 0 \\ 0 & b\end{bmatrix}; a, b \in R - \left\{ 0 \right\} \right\}\] defined by A * B = AB is a binary operation.
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 operation 'o' on Q defined by \[\text{a o b }= \frac{ab}{2}\] for all a, b ∈ Q ?
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 * 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 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 identity element in A ?
Let 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by \[a o b = \frac{ab}{2}, \text{for all a, b} \in Q_0\].
Show that 'o' is both commutative and associate ?
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.
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 ×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.
Write the inverse of 5 under multiplication modulo 11 on the set {1, 2, ... ,10}.
Let * be a binary operation on set of integers I, defined by a * b = 2a + b − 3. Find the value of 3 * 4.
On the power set P of a non-empty set A, we define an operation ∆ by
\[X ∆ Y = \left( \overline{X} \cap Y \right) \cup \left( X \cap \overline{Y} \right)\]
Then which are of the following statements is true about ∆.
Mark the correct alternative in the following question:-
For the binary operation * on Z defined by a * b = a + b + 1, the identity element is ________________ .
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 __________ .
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 __________ .
The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is ________________ .
Let * be a binary operation on R defined by a * b = ab + 1. Then, * is _________________ .
The law a + b = b + a is called _________________ .
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 _________________ .
For the multiplication of matrices as a binary operation on the set of all matrices of the form \[\begin{bmatrix}a & b \\ - b & a\end{bmatrix}\] a, b ∈ R the inverse of \[\begin{bmatrix}2 & 3 \\ - 3 & 2\end{bmatrix}\] is ___________________ .
Let * be a binary operation defined on Q+ by the rule
\[a * b = \frac{ab}{3} \text{ for all a, b } \in Q^+\] The inverse of 4 * 6 is ___________ .
Determine whether * is a binary operation on the sets-given below.
a * b – a.|b| on R
Determine whether * is a binary operation on the sets-given below.
a * b = min (a, b) on A = {1, 2, 3, 4, 5}
Choose the correct alternative:
Subtraction is not a binary operation in
Let * be a binary operation defined on Q. Find which of the following binary operations are associative
a * b = a – b for a, b ∈ Q
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative
a * b = a + ab ∀ a, b ∈ Q
Let * be a binary operation on Q, defined by a * b `= (3"ab")/5` is ____________.
Which of the following is not a binary operation on the indicated set?
