Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
(i) On Q, the operation * is defined as a * b = a − b.
It can be observed that:
`1/2 * 1/3 = 1/2 - 1/3 = (3-2)/6 = 1/6` " and " `1/3 * 1/2 = 1/3- 1/2 = (2-3)/6 = (-1)/6`
`:. 1/2 * 1/3 != 1/3 * 1/2` where `1/2, 1/3 in Q`
Thus, the operation * is not commutative.
It can also be observed that:
`(1/2 * 1/3) * 1/4 = (1/2 - 1/3) * 1/4 = 1/6 * 1/4 = 1/6 - 1/4 = (2 -3)/12 = (-1)/12`
`1/2 * (1/3 * 1/4) = 1/2 * (1/3 - 1/4) = 1/2 * 1/12 = 1/2 - 1/12 = (6 -1)/12 = 5/12`
`:. (1/2 * 1/3) * 1/4 != 1/2 * (1/3 * 1/4)` where `1/2, 1/3, 1/4 in Q`
Thus, the operation * is not associative.
(ii) On Q, the operation * is defined as a * b = a2 + b2.
For a, b ∈ Q, we have:
`a* b = a^2 + b^2 = b^2 + a^2 = b * a`
∴a * b = b * a
Thus, the operation * is commutative.
It can be observed that:
(1*2)*3 =(12 + 22)*3 = (1+4)*3 = 5*3 = 52+32= 25 + 9 = 34
1*(2*3)=1*(22+32) = 1*(4+9) = 1*13 = 12+ 132 = 1 + 169 = 170
∴ (1*2)*3 ≠ 1*(2*3) , where 1, 2, 3 ∈ Q
Thus, the operation * is not associative.
(iii) On Q, the operation * is defined as a * b = a + ab.
It can be observed that:
`1 *2 = 1 + 1 xx 2 = 1 + 2 = 3`
`2 * 1 = 2 + 2 xx 1 = 2 + 2 =4`
`:. 1 * 2 != 2 *1 where 1, 2 in Q`
Thus, the operation * is not commutative.
It can also be observed that:
`(1 * 2)*3 = (1 + 1 xx 2) * 3 = 3 * 3 = 3 + 3 xx 3 = 3 + 9 = 12`
`1 * (2 * 3) = 1 *(2+2xx3) = 1 + 1 xx 8 = 9`
`:.(1 * 2)*3 != 1 *(2 * 3)` where 1, 2. 3 ∈ Q
Thus, the operation * is not associative.
(iv) On Q, the operation * is defined by a * b = (a − b)2.
For a, b ∈ Q, we have:
a * b = (a − b)2
b * a = (b − a)2 = [− (a − b)]2 = (a − b)2
∴ a * b = b * a
Thus, the operation * is commutative.
It can be observed that:
`(1 * 2)*3 = (1 - 2)^2 * 3 = (-1)^2 * 3 = 1 * 3 = (1-3)^2 = (-2)^2 = 4`
`1 * (2 * 3) = 1 * (2 - 3)^2 = 1*(-1)^2 = 1*1 = (1 - 1)^2 = 0`
`:. (1 * 2) * 3 != 1 *(2 * 3)` where 1,2,3 ∈ Q
Thus, the operation * is not associative.
On Q, the operation * is defined as `a * b = "ab"/4`
For a, b ∈ Q, we have:
`a * b = "ab"/4 = ba/4 = b * a`
∴ a * b = b * a
Thus, the operation * is commutative.
For a, b, c ∈ Q, we have:
`(a * b) * c = "ab"/4 * c = ("ab"/4 . c)/4 = "abc"/16`
`a * (b * c) = a * bc/4= (a . "bc"/4)/4 = "abc"/16`
= ∴(a * b) * c = a * (b * c)
Thus, the operation * is associative.
(vi) On Q, the operation * is defined as a * b = ab2
It can be observed that:
`1/2 * 1/3 = 1/2 . (1/3)^2 = 1/2 . 1/9 = 1/18`
`1/3 * 1/2 = 1/3 . (1/2)^2 = 1/3 . 1/4 = 1/12`
`:. 1/2 * 1/3 != 1/3 * 1/2` where `1/2, 1/3 in Q`
Thus, the operation * is not commutative.
It can also be observed that:
`(1/2 * 1/3) * 1/4 = [1/2.(1/3)^2]* 1/4 = 1/18 * 1/4 = 1/18 . (1/4)^2 = 1/(18xx16)`
`1/2 * (1/3 * 1/4) = 1/2 * [1/3 . (1/4)^2] = 1/2 * 1/48 = 1/2 . (1/48)^2 = 1/(2 xx (48)^2)`
`:. (1/2 * 1/3) * 1/4 != 1/2 (1/3 * 1/4)` where `1/2, 1/3, 1.4 in Q`
Thus, the operation * is not associative.
Hence, the operations defined in (ii), (iv), (v) are commutative and the operation defined in (v) is associative.
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
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
Find which of the operations given above has identity.
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.
Consider a binary operation * on N defined as a * b = a3 + b3. Choose the correct answer.
(A) Is * both associative and commutative?
(B) Is * commutative but not associative?
(C) Is * associative but not commutative?
(D) Is * neither commutative nor associative?
Given a non-empty set X, let *: P(X) × P(X) → P(X) be defined as A * B = (A − B) ∪ (B −A), &mnForE; A, B ∈ P(X). Show that the empty set Φ is the identity for the operation * and all the elements A of P(X) are invertible with A−1 = A. (Hint: (A − Φ) ∪ (Φ − A) = Aand (A − A) ∪ (A − A) = A * A = Φ).
Define a binary operation *on the set {0, 1, 2, 3, 4, 5} as
a * b = `{(a+b, "if a+b < 6"), (a + b - 6, if a +b >= 6):}`
Show that zero is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 − a being the inverse of a.
Discuss the commutativity and associativity of binary operation '*' defined on A = Q − {1} by the rule a * b= a − b + ab for all, a, b ∊ A. Also find the identity element of * in A and hence find the invertible elements of A.
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.
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.
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 N defined by a * b = gcd(a, b) for all a, b ∈ N ?
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.
Let * be a binary operation on Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element ?
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 ?
For the binary operation ×7 on the set S = {1, 2, 3, 4, 5, 6}, compute 3−1 ×7 4.
Write the identity element for the binary operation * on the set R0 of all non-zero real numbers by the rule \[a * b = \frac{ab}{2}\] for all a, b ∈ R0.
Define an associative binary operation on a set.
A binary operation * is defined on the set R of all real numbers by the rule \[a * b = \sqrt{ a^2 + b^2} \text{for all a, b } \in R .\]
Write the identity element for * on R.
On the power set P of a non-empty set A, we define an operation ∆ by
\[X ∆ Y = \left( \overline{X} \cap Y \right) \cup \left( X \cap \overline{Y} \right)\]
Then which are of the following statements is true about ∆.
If G is the set of all matrices of the form
\[\begin{bmatrix}x & x \\ x & x\end{bmatrix}, \text{where x } \in R - \left\{ 0 \right\}\] then the identity element with respect to the multiplication of matrices as binary operation, is ______________ .
Let * be a binary operation defined on set Q − {1} by the rule a * b = a + b − ab. Then, the identify element for * is ____________ .
On Z an operation * is defined by a * b = a2 + b2 for all a, b ∈ Z. The operation * on Z is _______________ .
Let * be a binary operation defined on Q+ by the rule
\[a * b = \frac{ab}{3} \text{ for all a, b } \in Q^+\] The inverse of 4 * 6 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 v B
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 A be Q\{1} Define * on A by x * y = x + y – xy. Is * binary on A? If so, examine the commutative and associative properties satisfied by * on A
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
In the set N of natural numbers, define the binary operation * by m * n = g.c.d (m, n), m, n ∈ N. Is the operation * commutative and associative?
Let R be the set of real numbers and * be the binary operation defined on R as a * b = a + b – ab ∀ a, b ∈ R. Then, the identity element with respect to the binary operation * is ______.
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative
a * b = a2 + b2 ∀ 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
Let A = N x N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Then * 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?
