Advertisements
Advertisements
प्रश्न
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
उत्तर
R.H.S. = (c * b) * a
= (b * c) * a [* is commutative]
= a * (b * c) [Again, as * is commutative]
= L.H.S.
∴ a * (b * c) = (c * b) * a
Therefore, statement (ii) is true.
APPEARS IN
संबंधित प्रश्न
Determine whether or not 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 R, define * by a * b = ab2
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.
Let * be a binary operation on the set Q of rational numbers as follows:
(i) a * b = a − b
(ii) a * b = a2 + b2
(iii) a * b = a + ab
(iv) a * b = (a − b)2
(v) a * b = ab/4
(vi) a * b = ab2
Find which of the binary operations are commutative and which are associative.
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.
Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.
Find the total number of binary operations on {a, b}.
Check the commutativity and associativity of the following binary operations '*'. on Q defined by a * b = a − b for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on Q defined by \[a * b = \frac{ab}{4}\] for all a, b ∈ Q ?
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.
The binary operation * is defined by \[a * b = \frac{ab}{7}\] on the set Q of all rational numbers. Show that * is associative.
Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Find the identity element in Q − {−1} ?
Let 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by \[a o b = \frac{ab}{2}, \text{for all a, b} \in Q_0\].
Show that 'o' is both commutative and associate ?
Construct the composition table for +5 on set S = {0, 1, 2, 3, 4}.
Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}.
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.
Define a commutative binary operation on a set.
Write the total number of binary operations on a set consisting of two elements.
Write the composition table for the binary operation ×5 (multiplication modulo 5) on the set S = {0, 1, 2, 3, 4}.
If a * b = a2 + b2, then the value of (4 * 5) * 3 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 __________ .
Subtraction of integers is ___________________ .
The law a + b = b + a is called _________________ .
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 ____________ .
Consider the binary operation * defined on the set A = {a, b, c, d} by the following table:
| * | a | b | c | d |
| a | a | c | b | d |
| b | d | a | b | c |
| c | c | d | a | a |
| d | d | b | a | c |
Is it commutative and associative?
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 M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}` and let * be the matrix multiplication. Determine whether M is closed under *. If so, examine the commutative and associative properties satisfied by * on M
Choose the correct alternative:
A binary operation on a set S is a function from
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 ______.
A binary operation on a set has always the identity element.
The binary operation * defined on set R, given by a * b `= "a+b"/2` for all a, b ∈ R 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 ____________.
