Advertisements
Advertisements
Question
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.
Solution
\[a, b \in Z^+ \]
\[ \Rightarrow a \in Z^+ \]
\[ \Rightarrow a * b \in Z^+ \]
\[\text{Therefore},\]
\[a * b \in Z^+ , \forall a, b \in Z^+ \]
\[Thus, * \text{ is a binary operation on } Z^+ .\]
APPEARS IN
RELATED QUESTIONS
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
For each binary operation * defined below, determine whether * is commutative or associative.
On Z, define a * b = a − b
For each binary operation * defined below, determine whether * is commutative or associative.
On Z+, define a * b = ab
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.
Determine whether the following operation define a binary operation on the given set or not :
\[' * ' \text{on Q defined by } a * b = \frac{a - 1}{b + 1} \text{for all a, b} \in Q .\]
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 = ab
Here, Z+ denotes the set of all non-negative integers.
Determine which of the following binary operation is associative and which is commutative : * on N defined by a * b = 1 for all a, b ∈ N ?
Check the commutativity and associativity of the following binary operations '*'. on N defined by a * b = 2ab for all a, b ∈ N ?
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 = ab + 1 for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on N, defined by a * b = ab for all a, b ∈ N ?
Check the commutativity and associativity of the following binary operation '*' on Z defined by a * b = a − b for all a, b ∈ Z ?
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 the set Q of all ration numbers if a binary operation * is defined by \[a * b = \frac{ab}{5}\] , prove that * is associative on Q.
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.
Construct the composition table for ×4 on set S = {0, 1, 2, 3}.
Find the inverse of 5 under multiplication modulo 11 on Z11.
Write the multiplication table for the set of integers modulo 5.
Consider the binary operation 'o' defined by the following tables on set S = {a, b, c, d}.
| o | a | b | c | d |
| a | a | a | a | a |
| b | a | b | c | d |
| c | a | c | d | b |
| d | a | d | b | c |
Show that the binary operation is commutative and associative. Write down the identities and list the inverse of elements.
Write the composition table for the binary operation multiplication modulo 10 (×10) on the set S = {2, 4, 6, 8}.
For the binary operation multiplication modulo 5 (×5) defined on the set S = {1, 2, 3, 4}. Write the value of \[\left( 3 \times_5 4^{- 1} \right)^{- 1}.\]
Subtraction of integers 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 ___________________ .
Let '*' be a binary operation on N defined by
a * b = 1.c.m. (a, b) for all a, b ∈ N
Find 2 * 4, 3 * 5, 1 * 6.
Examine whether the operation *defined on R by a * b = ab + 1 is (i) a binary or not. (ii) if a binary operation, is it associative or not?
On Z, define * by (m * n) = mn + nm : ∀m, n ∈ Z Is * binary on Z?
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?
Choose the correct alternative:
A binary operation on a set S is a function from
Choose the correct alternative:
In the set R of real numbers ‘*’ is defined as follows. Which one of the following is not a binary operation on R?
Choose the correct alternative:
In the set Q define a ⨀ b = a + b + ab. For what value of y, 3 ⨀ (y ⨀ 5) = 7?
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative
a * b = (a – b)2 ∀ a, b ∈ Q
Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.
Let * be a binary operation on set Q – {1} defind by a * b = a + b – ab : a, b ∈ Q – {1}. Then * 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.
If * is a binary operation on the set of integers I defined by a * b = 3a + 4b - 2, then find the value of 4 * 5.
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`.
