Advertisements
Advertisements
Question
Check the commutativity and associativity of the following binary operations '*'. on N defined by a * b = 2ab for all a, b ∈ N ?
Solution
Commutativity:
\[\text{Let } a, b \in N . \text{Then}, \]
\[a * b = 2^{ab} \]
\[ = 2^{ba} \]
\[ = b * a\]
\[\text{Therefore},\]
\[a * b = b * a, \forall a, b \in N\]
Thus, * is commutative on N.
Associativity:
\[\text{Let a}, b, c \in N . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( 2^{bc} \right)\]
\[ = 2^{a * 2^{bc}} \]
\[\left( a * b \right) * c = \left( 2^{ab} \right) * c\]
\[ = 2^{ab * 2^c} \]
\[\text{Therefore},\]
\[a * \left( b * c \right) \neq \left( a * b \right) * c\]
Thus, * is not associative on N.
APPEARS IN
RELATED QUESTIONS
LetA= R × 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 identity element for * on A. Also find the inverse of every element (a, b) ε A.
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
For each binary operation * defined below, determine whether * is commutative or associative.
On Z+, define a * b = ab
Determine whether the following operation define a binary operation on the given set or not : '×6' on S = {1, 2, 3, 4, 5} defined by
a ×6 b = Remainder when ab is divided by 6.
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.
Let * be a binary operation on the set I of integers, defined by a * b = 2a + b − 3. Find the value of 3 * 4.
Find the total number of binary operations on {a, b}.
Let S be the set of all rational numbers of the form \[\frac{m}{n}\] , where m ∈ Z and n = 1, 2, 3. Prove that * on S defined by a * b = ab is not a binary operation.
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.
Check the commutativity and associativity of the following binary operation '*'. on Z defined by a * b = a + b + ab for all a, b ∈ Z ?
On Q, the set of all rational numbers, * is defined by \[a * b = \frac{a - b}{2}\] , shown that * is no associative ?
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.
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 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 ?
For the binary operation ×10 on set S = {1, 3, 7, 9}, find the inverse of 3.
Find the inverse of 5 under multiplication modulo 11 on Z11.
Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as \[a * b = \begin{cases}a + b & ,\text{ if a + b} < 6 \\ a + b - 6 & , \text{if a + b} \geq 6\end{cases}\]
Show that 0 is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 − a being the inverse of a.
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.
Let * be a binary operation on N given by a * b = HCF (a, b), a, b ∈ N. Write the value of 22 * 4.
If a * b denote the bigger among a and b and if a ⋅ b = (a * b) + 3, then 4.7 = __________ .
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 ∆.
If a binary operation * is defined on the set Z of integers as a * b = 3a − b, then the value of (2 * 3) * 4 is ___________ .
Which of the following is true ?
The number of binary operation that can be defined on a set of 2 elements is _________ .
If * is defined on the set R of all real numbers by *: a*b = `sqrt(a^2 + b^2 ) `, find the identity elements, if it exists in R with respect to * .
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 *
Let A = {a + `sqrt(5)`b : a, b ∈ Z}. Check whether the usual multiplication is a binary operation on A
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.
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
Choose the correct alternative:
A binary operation on a set S is a function from
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 * be the binary operation defined on Q. Find which of the following binary operations are commutative
a * b = a – b ∀ a, b ∈ Q
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative
a * b = a2 + b2 ∀ a, b ∈ Q
Let * be the binary operation on N given by a * b = HCF (a, b) where, a, b ∈ N. Find the value of 22 * 4.
Consider the binary operation * on Q defind by a * b = a + 12b + ab for a, b ∈ Q. Find 2 * `1/3`.
