Advertisements
Advertisements
Question
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.
Advertisements
Solution 1
(a, b) * (c, d) = (ac, b + ad)
(c, d) * (a, b) = (ca, d + cf)
Not commutative
(a, b) * [(c, d) * (e, f)]
= (a, b) * [ce, d + cf]
= [ace, b + ad + acf]
Now,
[(a, b) * (c, d)] * (e, f)
= [ac, b + ad] * (e, f)
= [ace, b + ad + acf]
∴ Associative
∵ (a, b) * [(c, d) * (e, f)]= [(a, b)] * [(c, d) * (e, f)]
1) (a, b) * e = (a, b)
⇒ a = ac
⇒ c = 1 and b = b + ad ⇒ ad = 0
⇒ d = 0
∴ (a, b) * (1, 0) = (a, b + a × 0) = (a, b)
⇒ (1,0) is identify
2) (a, b) * (c, d) = e = (1, 0)
⇒ ac = 1 and b + ad = 0
⇒ `d = (-b)/a`
∴ Inverse of element
∴ Inverse of element of a, b is `(1/a, (-b)/a)`
Solution 2
Let A=Q×Q and * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d)∈A.
Commutativity:
Let X = (a, b) and Y = (c, d) ∈ A,∀ a, c ∈ Q and b, d ∈ Q. Then,
X * Y =(ac, b + ad)
Y * X=(ca, d + cb)
Therefore, X * Y ≠ Y * X ∀ X,Y∈A
Thus, * is not commutative on A.
Associativity:
Let X = (a, b), Y = (c, d) and Z=( e, f),∀ a, c, e ∈ Q and b, d, f ∈ Q
X*(Y*Z)=(a, b)*(ce, d+cf)
=(ace, b + ad + acf)
(X * Y)* Z=(ac, b + ad)*(e,f)
= (ace, b + ad + acf)
∴ X*(Y * Z) = (X * Y)*Z, ∀ X, Y, Z ∈ A
Thus,* is associative on A.
1) Let E = (x, y) be the identity element in A with respect to *,∀ x ∈ Q and y ∈ Q such that
X * E = X = E * X,∀ X ∈ A
⇒ X * E = X and E * X = X
⇒(ax, b + ay)=(a, b) and (xa, y + xb) = (a, b)
Considering (ax, b+ay)=(a, b)
⇒ ax = a
⇒ x = 1 and b + ay = b
⇒ y = 0
Considering (xa, y + xb) = (a, b)
⇒ xa = a
⇒ x = 1 and y + xb = b
⇒y = 0 [∵ x=1]
∴ (1, 0) is the identity element in A with respect to *.
Let F = (m, n) be the inverse in A∀ m ∈ Q and n ∈ Q
X * F = E and F * X = E
⇒(am, b + an) = (1, 0) and (ma, n + mb) = (1, 0)
Considering (am, b + an)=(1, 0)
⇒ am = 1
⇒m = 1/a and b + an = 0
`=> n = (-b)/a`
Considering (ma, n+mb)=(1, 0)
⇒ ma = 1
`=> m = 1/a and n + mb = 0`
`=> n = (-b)/a` [∵ m = 1/a]
∴ The inverse of (a, b) ∈ A with respect to * is `(1/a, (-b)/a)`
RELATED QUESTIONS
Let A = Q ✕ Q, where Q is the set of all rational numbers, and * be a binary operation defined on A by (a, b) * (c, d) = (ac, b + ad), for all (a, b) (c, d) ∈ A.
Find
(i) the identity element in A
(ii) the invertible element of A.
(iii)and hence write the inverse of elements (5, 3) and (1/2,4)
For each binary operation * defined below, determine whether * is commutative or associative.
On Q, define a * b = `(ab)/2`
Let A = N × N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d)
Show that * is commutative and associative. Find the identity element for * on A, if any.
Determine whether the following operation define a binary operation on the given set or not :
\[' +_6 ' \text{on S} = \left\{ 0, 1, 2, 3, 4, 5 \right\} \text{defined by}\]
\[a +_6 b = \begin{cases}a + b & ,\text{ if a} + b < 6 \\ a + b - 6 & , \text{if a} + b \geq 6\end{cases}\]
Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.
Let S be the set of all rational numbers of the form \[\frac{m}{n}\] , where m ∈ Z and n = 1, 2, 3. Prove that * on S defined by a * b = ab is not a binary operation.
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 operation'*' on Q defined by a * b = (a − b)2 for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on Z defined by a * b = a + b − ab for all a, b ∈ Z ?
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.
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.
For the binary operation ×7 on the set S = {1, 2, 3, 4, 5, 6}, compute 3−1 ×7 4.
Write the inverse of 5 under multiplication modulo 11 on the set {1, 2, ... ,10}.
For the binary operation multiplication modulo 10 (×10) defined on the set S = {1, 3, 7, 9}, write the inverse of 3.
Let * be a binary operation on N given by a * b = HCF (a, b), a, b ∈ N. Write the value of 22 * 4.
Q+ denote the set of all positive rational numbers. If the binary operation a ⊙ on Q+ is defined as \[a \odot = \frac{ab}{2}\] ,then the inverse of 3 is __________ .
A binary operation * on Z defined by a * b = 3a + b for all a, b ∈ Z, is ________________ .
On the set Q+ of all positive rational numbers a binary operation * is defined by \[a * b = \frac{ab}{2} \text{ for all, a, b }\in Q^+\]. The inverse of 8 is _________ .
Determine whether * is a binary operation on the sets-given below.
a * b – a.|b| on R
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) v C
Choose the correct alternative:
Which one of the following is a binary operation on N?
Let * be a binary operation on Q, defined by a * b `= (3"ab")/5` is ____________.
Let * be a binary operation on set Q of rational numbers defined as a * b `= "ab"/5`. Write the identity for * ____________.
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.
Subtraction and division are not binary operation on.
