Advertisements
Advertisements
प्रश्न
On Z, define * by (m * n) = mn + nm : ∀m, n ∈ Z Is * binary on Z?
Advertisements
उत्तर
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
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 = ab
Let * be the binary operation on N given by a * b = L.C.M. of a and b. Find
(i) 5 * 7, 20 * 16
(ii) Is * commutative?
(iii) Is * associative?
(iv) Find the identity of * in N
(v) Which elements of N are invertible for the operation *?
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.
On the set Z of integers a binary operation * is defined by a * b = ab + 1 for all a , b ∈ Z. Prove that * is not associative on Z.
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 Q, the set of all rational numbers, * is defined by \[a * b = \frac{a - b}{2}\] , shown that * is no associative ?
Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.
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 Z defined by
a * b = a + b − 4 for all a, b ∈ Z Find the identity element in Z ?
Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Find the invertible elements in Z ?
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.
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.
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 ________________ .
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 * be binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R. Then the operation * is ______.
Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.
Which of the following is not a binary operation on the indicated set?
a * b = `((a + b))/2` ∀a, b ∈ N is
