Advertisements
Advertisements
Question
On Z, define * by (m * n) = mn + nm : ∀m, n ∈ Z Is * binary on Z?
Advertisements
Solution
No.
* is not a binary operation on Z.
Reason: Since m, n ∈ Z.
So, m, n can be negative also.
Now, if n is negative (Le.) say n = – k where k is +ve.
Then mn = m–k = `1/"m"^"k"` ∈ Z.
Similarly, when m is negative then nm ∉ Z.
∴ m * n ∉ Z.
⇒ * is not a binary operation on Z.
APPEARS IN
RELATED QUESTIONS
Let * be a binary operation, on the set of all non-zero real numbers, given by `a** b = (ab)/5` for all a,b∈ R-{0} that 2*(x*5)=10
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|
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 Q, define a * b = ab + 1
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.
Check the commutativity and associativity of the following binary operations '⊙' on Q defined by a ⊙ b = a2 + b2 for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on N defined by a * b = gcd(a, b) for all a, b ∈ N ?
Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows (a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 × R :
Find the invertible elements in A ?
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 ?
Find the inverse of 5 under multiplication modulo 11 on Z11.
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.
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 identity element for the binary operation * defined on the set R of all real numbers by the rule
\[a * b = \frac{3ab}{7} \text{ for all a, b} \in R .\] ?
Let * be a binary operation, on the set of all non-zero real numbers, given by \[a * b = \frac{ab}{5} \text { for all a, b } \in R - \left\{ 0 \right\}\]
Write the value of x given by 2 * (x * 5) = 10.
Let * be a binary operation on N given by a * b = HCF (a, b), a, b ∈ N. Write the value of 22 * 4.
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 be Q\{1}. Define * on A by x * y = x + y – xy. Is * binary on A? If so, examine the existence of an identity, the existence of inverse properties for the operation * on A
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 ∀ a, b ∈ Q
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative
a * b = a + ab ∀ a, b ∈ Q
