Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
If a = 1 and b = 2 in Z+, then
\[a * b = a - b\]
\[ = 1 - 2\]
\[ = - 1 \not\in Z^+ \left[ \because Z^+ \text{ is the set of non-negative integers } \right]\]
\[ \Rightarrow \text{Fora} = 1 \text{ and }b = 2, \]
\[a * b \not\in Z^+ \]
\[\text{Thus}, * \text{ is not a binary operation on } Z^+ .\]
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
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|
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 = a − b
State whether the following statements are true or false. Justify.
If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a
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 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.
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.
Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative ?
The binary operation * is defined by \[a * b = \frac{ab}{7}\] on the set Q of all rational numbers. Show that * is associative.
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 Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that '*' is both commutative and associative on Q − {−1}.
Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element ?
Let 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by \[a o b = \frac{ab}{2}, \text{ for all a, b } \in Q_0\]:
Find the invertible elements of Q0 ?
Construct the composition table for ×4 on set S = {0, 1, 2, 3}.
For the binary operation ×10 on set S = {1, 3, 7, 9}, find the inverse of 3.
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.
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 defined by a * b = 3a + 4b − 2. Find 4 * 5.
Which of the following is true ?
The law a + b = b + a is called _________________ .
On Z an operation * is defined by a * b = a2 + b2 for all a, b ∈ Z. The operation * on Z is _______________ .
For the binary operation * defined on R − {1} by the rule a * b = a + b + ab for all a, b ∈ R − {1}, the inverse of a is ________________ .
The number of commutative binary operations that can be defined on a set of 2 elements is ____________ .
Consider the binary operation * defined by the following tables on set S = {a, b, c, d}.
| * | a | b | c | d |
| a | a | b | c | d |
| b | b | a | d | c |
| c | c | d | a | b |
| d | d | c | b | a |
Show that the binary operation is commutative and associative. Write down the identities and list the inverse of elements.
Let A = {a + `sqrt(5)`b : a, b ∈ Z}. Check whether the usual multiplication is a binary operation on A
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
Choose the correct alternative:
Subtraction is not a binary operation in
Is the binary operation * defined on Z (set of integer) by m * n = m – n + mn ∀ m, n ∈ Z commutative?
Let * be a binary operation defined on Q. Find which of the following binary operations are associative
a * b = a – b + ab for a, b ∈ Q
Let * be a binary operation defined on Q. Find which of the following binary operations are associative
a * b = ab2 for a, b ∈ Q
The identity element for the binary operation * defined on Q ~ {0} as a * b = `"ab"/2` ∀ a, b ∈ Q ~ {0} is ______.
The binary operation * defined on set R, given by a * b `= "a+b"/2` for all a, b ∈ R is ____________.
Let * be a binary operation on the set of integers I, defined by a * b = a + b – 3, then find the value of 3 * 4.
a * b = `((a + b))/2` ∀a, b ∈ N is
