Advertisements
Advertisements
प्रश्न
Let * be a binary operation on R defined by a * b = ab + 1. Then, * is _________________ .
पर्याय
commutative but not associative
associative but not commutative
neither commutative nor associative
both commutative and associative
Advertisements
उत्तर
commutative but not associative
Commutativity:
\[\text { Let } a, b \in R\]
\[a * b = ab + 1\]
\[ = ba + 1\]
\[ = b * a\]
\[\text { Therefore },\]
\[a * b = b * a, \forall a, b \in R\]
Therefore, * is commutative on R.
Associativity:
\[\text{ Let }a, b, c \in R\]
\[a * \left( b * c \right) = a * \left( bc + 1 \right)\]
\[ = a\left( bc + 1 \right) + 1\]
\[ = abc + a + 1\]
\[\left( a * b \right) * c = \left( ab + 1 \right) * c\]
\[ = \left( ab + 1 \right)c + 1\]
\[ = abc + c + 1\]
\[\therefore a * \left( b * c \right) \neq \left( a * b \right) * c\]
\[\text{ For example }:a=1,b = 2 \text{ and } c = 3 \left[ \text{ which belong to R } \right]\]
\[\text{ Now }, \]
\[1 * \left( 2 * 3 \right) = 1 * \left( 6 + 1 \right)\]
\[ = 1 * 7\]
\[ = 7 + 1\]
\[ = 8\]
\[\left( 1 * 2 \right) * 3 = \left( 2 + 1 \right) * 3\]
\[ = 3 * 3\]
\[ = 9 + 1\]
\[ = 10\]
\[ \Rightarrow 1 * \left( 2 * 3 \right) \neq \left( 1 * 2 \right) * 3\]
\[\text { Therefore }, \exists a=1,b = 2 \text{ and } c = 3 \text{ which belong to R such that a } * \left( b * c \right) \neq \left( a * b \right) * c\]
Hence, * is not associative on R.
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
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
Let*′ be the binary operation on the set {1, 2, 3, 4, 5} defined by a *′ b = H.C.F. of a and b. Is the operation *′ same as the operation * defined in Exercise 4 above? Justify your answer.
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*.
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.
Let S = {a, b, c}. Find the total number of binary operations on S.
Let A be any set containing more than one element. Let '*' be a binary operation on A defined by a * b = b for all a, b ∈ A Is '*' commutative or associative on A ?
Check the commutativity and associativity of the following binary operation '*' on Z defined by a * b = a + b − ab 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 ?
On Q, the set of all rational numbers, * is defined by \[a * b = \frac{a - b}{2}\] , shown that * is no associative ?
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 :
Show that '⊙' is commutative and associative on A ?
Let R0 denote the set of all non-zero real numbers and let A = R0 × R0. If '*' is a binary operation on A defined by
(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A
Show that '*' is both commutative and associative on A ?
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 ×5 on Z5 = {0, 1, 2, 3, 4}.
Find the inverse of 5 under multiplication modulo 11 on Z11.
Define identity element for a binary operation defined on a set.
Let * be a binary operation on set of integers I, defined by a * b = 2a + b − 3. Find the value of 3 * 4.
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 ______________ .
Let * be a binary operation defined on set Q − {1} by the rule a * b = a + b − ab. Then, the identify element for * is ____________ .
The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is ________________ .
The binary operation * is defined by a * b = a2 + b2 + ab + 1, then (2 * 3) * 2 is equal to ______________ .
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 number by *: a * b = `sqrt(a^2 + b^2)` find the identity element if exist in R with respect to *
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?
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.
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.
Fill in the following table so that the binary operation * on A = {a, b, c} is commutative.
| * | a | b | c |
| a | b | ||
| b | c | b | a |
| c | a | c |
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
Choose the correct alternative:
Which one of the following is a binary operation on N?
Choose the correct alternative:
If a * b = `sqrt("a"^2 + "b"^2)` on the real numbers then * is
Let R be the set of real numbers and * be the binary operation defined on R as a * b = a + b – ab ∀ a, b ∈ R. Then, the identity element with respect to the binary operation * is ______.
Let N be the set of natural numbers. Then, the binary operation * in N defined as a * b = a + b, ∀ a, b ∈ N has identity element.
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 ______.
Consider the binary operation * on Q defind by a * b = a + 12b + ab for a, b ∈ Q. Find 2 * `1/3`.
