Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Commutativity:
\[\text{Let a}, b \in Z . \text{Then}, \]
\[a * b = a + 3b - 4\]
\[b * a = b + 3a - 4\]
\[a * b \neq b * a\]
\[\text{Let }a = 1, b = 2\]
\[1 * 2 = 1 + 6 - 4\]
\[ = 3\]
\[2 * 1 = 2 + 3 - 4\]
\[ = 1\]
\[\text{Therefore}, \exists \text{ a} = 1, b = 2 \in \text{Z such that a} * b \neq b * a\]
Thus, * is not commutative on Z.
Associativity:
\[\text{Let a}, b, c \in Z . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( b + 3c - 4 \right)\]
\[ = a + 3\left( b + 3c - 4 \right) - 4\]
\[ = a + 3b + 9c - 12 - 4\]
\[ = a + 3b + 9c - 16\]
\[\left( a * b \right) * c = \left( a + 3b - 4 \right) * c\]
\[ = a + 3b - 4 + 3c - 4\]
\[ = a + 3b + 3c - 8\]
\[\text{Thus, a} * \left( b * c \right) \neq \left( a * b \right) * c\]
\[\text{ If a } = 1, b = 2, c = 3\]
\[1 * \left( 2 * 3 \right) = 1 * \left( 2 + 9 - 4 \right)\] \[ = 1 * 7 \]
\[ = 1 + 21 - 4\]
\[ = 18\]
\[\left( 1 * 2 \right) * 3 = \left( 1 + 6 - 4 \right) * 3\]
\[ = 3 * 3\]
\[ = 3 + 9 - 4\]
\[ = 8\]
\[\text{Therefore}, \exists \text{ a} = 1, b = 2, c = 3 \in \text{Z such that a } * \left( b * c \right) \neq \left( a * b \right) * c\]
Thus, * is not associative on Z.
APPEARS IN
संबंधित प्रश्न
Show that the binary operation * on A = R – { – 1} defined as a*b = a + b + ab for all a, b ∈ A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible.
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 − b|
Consider the binary operation ∨ on the set {1, 2, 3, 4, 5} defined by a ∨b = min {a, b}. Write the operation table of the operation∨.
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.
Is * defined on the set {1, 2, 3, 4, 5} by a * b = L.C.M. of a and b a binary operation? Justify your answer.
Find which of the operations given above has identity.
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.
Let A = Q x Q and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d) ∈ A. Determine, whether * is commutative and associative. Then, with respect to * on A
1) Find the identity element in A
2) Find the invertible elements of A.
If a * b denotes the larger of 'a' and 'b' and if a∘b = (a * b) + 3, then write the value of (5) ∘ (10), where * and ∘ are binary operations.
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 = ab2
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.
The binary operation * : R × R → R is defined as a * b = 2a + b. Find (2 * 3) * 4.
Let '*' be a binary operation on N defined by a * b = 1.c.m. (a, b) for all a, b ∈ N
Check the commutativity and associativity of '*' on N.
Determine which of the following binary operations are associative and which are commutative : * on Q defined by \[a * b = \frac{a + b}{2} \text{ for all a, b } \in Q\] ?
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 '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 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 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 ?
If the binary operation * on the set Z is defined by a * b = a + b −5, the find the identity element with respect to *.
Find the inverse of 5 under multiplication modulo 11 on Z11.
On the set Z of all integers a binary operation * is defined by a * b = a + b + 2 for all a, b ∈ Z. Write the inverse of 4.
Define an associative binary operation on a set.
Write the identity element for the binary operation * defined on the set R of all real numbers by the rule
\[a * b = \frac{3ab}{7} \text{ for all a, b} \in R .\] ?
Let * be a binary operation, on the set of all non-zero real numbers, given by \[a * b = \frac{ab}{5} \text { for all a, b } \in R - \left\{ 0 \right\}\]
Write the value of x given by 2 * (x * 5) = 10.
Let * be a binary operation defined by a * b = 3a + 4b − 2. Find 4 * 5.
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 ____________ .
Q+ is the set of all positive rational numbers with the binary operation * defined by \[a * b = \frac{ab}{2}\] for all a, b ∈ Q+. The inverse of an element a ∈ Q+ 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 __________ .
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 _________ .
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 v B
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 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) v C
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
Choose the correct alternative:
Which one of the following is a binary operation on N?
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 ____________.
Which of the following is not a binary operation on the indicated set?
A binary operation A × A → is said to be associative if:-
a * b = `((a + b))/2` ∀a, b ∈ N is
