Advertisements
Advertisements
Question
Determine whether the following operation define a binary operation on the given set or not : 'O' on Z defined by a O b = ab for all a, b ∈ Z.
Solution
Both a = 3 and b = -1 belong to Z.
⇒ a * b = 3-1
=`1/3` ∉ Z
Thus, * is not a binary operation on Z.
APPEARS IN
RELATED QUESTIONS
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
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
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.
Given a non-empty set X, let *: P(X) × P(X) → P(X) be defined as A * B = (A − B) ∪ (B −A), &mnForE; A, B ∈ P(X). Show that the empty set Φ is the identity for the operation * and all the elements A of P(X) are invertible with A−1 = A. (Hint: (A − Φ) ∪ (Φ − A) = Aand (A − A) ∪ (A − A) = A * A = Φ).
Determine whether the following operation define a binary operation on the given set or not : '*' on N defined by a * b = a + b - 2 for all a, b ∈ N
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 the following operation define a binary operation on the given set or not :
\[' +_6 ' \text{on S} = \left\{ 0, 1, 2, 3, 4, 5 \right\} \text{defined by}\]
\[a +_6 b = \begin{cases}a + b & ,\text{ if a} + b < 6 \\ a + b - 6 & , \text{if a} + b \geq 6\end{cases}\]
Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.
Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = a + ab 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 the set Z of integers, if the binary operation * is defined by a * b = a + b + 2, then find the identity element.
Let * be a binary operation on Q0 (set of non-zero rational numbers) defined by \[a * b = \frac{ab}{5} \text{for all a, b} \in Q_0\]
Show that * is commutative as well as associative. Also, find its identity element if it exists.
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
Find the invertible element in A ?
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 ×4 on set S = {0, 1, 2, 3}.
Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}.
Construct the composition table for ×5 on Z5 = {0, 1, 2, 3, 4}.
Write the identity element for the binary operation * on the set R0 of all non-zero real numbers by the rule \[a * b = \frac{ab}{2}\] for all a, b ∈ R0.
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.
Write the composition table for the binary operation ×5 (multiplication modulo 5) on the set S = {0, 1, 2, 3, 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 binary operation * is defined on the set Z of integers as a * b = 3a − b, then the value of (2 * 3) * 4 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 ______________ .
Consider the binary operation * defined on Q − {1} by the rule
a * b = a + b − ab for all a, b ∈ Q − {1}
The identity element in Q − {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 ___________________ .
The number of commutative binary operations that can be defined on a set of 2 elements 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 v B) ∧ C
Choose the correct alternative:
Which one of the following is a binary operation on N?
Let * be binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R. Then the operation * is ______.
Let A = N x N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Then * is ____________.
Let * be the binary operation on N given by a * b = HCF (a, b) where, a, b ∈ N. Find the value of 22 * 4.
a * b = `((a + b))/2` ∀a, b ∈ N is
