Advertisements
Advertisements
प्रश्न
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 commutative as well as associative ?
Advertisements
उत्तर
Commutativity :
\[\text{Let }a, b \in S . \text{Then}, \]
\[a * b = a + b - ab\]
\[ = b + a - ba\]
\[ = b * a\]
\[\text{Therefore}, \]
\[a * b = b * a, \forall a, b \in S\]
Thus, * is commutative on S.
Associativity:
\[\text{Let} a, b, c \in S . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( b + c - bc \right)\]
\[ = a + b + c - bc - a\left( b + c - bc \right)\]
\[ = a + b + c - bc - ab - ac + abc\]
\[\left( a * b \right) * c = \left( a + b - ab \right) * c\]
\[ = a + b - ab + c - \left( a + b - ab \right)c\]
\[ = a + b + c - ab - ac - bc + abc\]
\[\text{Therefore}, \]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in S\]
Thus , * is associative on S.
So, * is commutative as well as associative.
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 R, define * by a * b = ab2
State whether the following statements are true or false. Justify.
For an arbitrary binary operation * on a set N, a * a = ∀ a a * 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 = 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 Z+, define * by a * b = a
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 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 operations '*'. on N defined by a * b = 2ab for all a, b ∈ N ?
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 '*' 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 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 A \[=\] R \[\times\] R and \[*\] be a binary operation on A defined by \[(a, b) * (c, d) = (a + c, b + d) .\] . Show that \[*\] is commutative and associative. Find the binary element for \[*\] on A, if any.
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.
For the binary operation multiplication modulo 10 (×10) defined on the set S = {1, 3, 7, 9}, write the inverse of 3.
For the binary operation multiplication modulo 5 (×5) defined on the set S = {1, 2, 3, 4}. Write the value of \[\left( 3 \times_5 4^{- 1} \right)^{- 1}.\]
Write the composition table for the binary operation ×5 (multiplication modulo 5) on the set S = {0, 1, 2, 3, 4}.
Let +6 (addition modulo 6) be a binary operation on S = {0, 1, 2, 3, 4, 5}. Write the value of \[2 +_6 4^{- 1} +_6 3^{- 1} .\]
If a * b = a2 + b2, then the value of (4 * 5) * 3 is _____________ .
If the binary operation * on Z is defined by a * b = a2 − b2 + ab + 4, then value of (2 * 3) * 4 is ____________ .
The binary operation * is defined by a * b = a2 + b2 + ab + 1, then (2 * 3) * 2 is equal to ______________ .
The law a + b = b + a is called _________________ .
A binary operation * on Z defined by a * b = 3a + b for all a, b ∈ Z, is ________________ .
Let * be a binary operation on N defined by a * b = a + b + 10 for all a, b ∈ N. The identity element for * in N is _____________ .
The number of commutative binary operations that can be defined on a set of 2 elements is ____________ .
Define an operation * on Q as follows: a * b = `(("a" + "b")/2)`; a, b ∈ Q. Examine the existence of identity and the existence of inverse for the operation * 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 |
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
Let M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}` and let * be the matrix multiplication. Determine whether M is closed under * . If so, examine the existence of identity, existence of inverse properties for the operation * on M
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
Choose the correct alternative:
If a * b = `sqrt("a"^2 + "b"^2)` on the real numbers then * is
In the set N of natural numbers, define the binary operation * by m * n = g.c.d (m, n), m, n ∈ N. Is the operation * commutative and associative?
Let A = N x N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Then * is ____________.
The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is ____________.
Subtraction and division are not binary operation on.
