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 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 R, define * by a * b = ab2
For each binary operation * defined below, determine whether * is commutative or associative.
On Q, define a * b = ab + 1
For each binary operation * defined below, determine whether * is commutative or associative.
On Q, define a * b = `(ab)/2`
Consider a binary operation * on the set {1, 2, 3, 4, 5} given by the following multiplication table.
(i) Compute (2 * 3) * 4 and 2 * (3 * 4)
(ii) Is * commutative?
(iii) Compute (2 * 3) * (4 * 5).
(Hint: use the following table)
| * | 1 | 2 | 3 | 4 | 5 |
| 1 | 1 | 1 | 1 | 1 | 1 |
| 2 | 1 | 2 | 1 | 2 | 1 |
| 3 | 1 | 1 | 3 | 1 | 1 |
| 4 | 1 | 2 | 1 | 4 | 1 |
| 5 | 1 | 1 | 1 | 1 | 5 |
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).
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 = a − b
Here, Z+ denotes the set of all non-negative integers.
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 R, define by a*b = ab2
Here, Z+ denotes the set of all non-negative integers.
Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.
Check the commutativity and associativity of the following binary operation '*' on Q defined by \[a * b = \frac{ab}{4}\] for all a, b ∈ Q ?
On the set Z of integers, if the binary operation * is defined by a * b = a + b + 2, then find the identity element.
On R − {1}, a binary operation * is defined by a * b = a + b − ab. Prove that * is commutative and associative. Find the identity element for * on R − {1}. Also, prove that every element of R − {1} is invertible.
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.
Construct the composition table for ×6 on set S = {0, 1, 2, 3, 4, 5}.
Find the inverse of 5 under multiplication modulo 11 on Z11.
Define a commutative binary operation on a set.
Define identity element for a binary operation defined on a set.
For the binary operation multiplication modulo 10 (×10) defined on the set S = {1, 3, 7, 9}, write the inverse of 3.
Write the composition table for the binary operation ×5 (multiplication modulo 5) on the set S = {0, 1, 2, 3, 4}.
Q+ is the set of all positive rational numbers with the binary operation * defined by \[a * b = \frac{ab}{2}\] for all a, b ∈ Q+. The inverse of an element a ∈ Q+ is ______________ .
If the binary operation ⊙ is defined on the set Q+ of all positive rational numbers by \[a \odot b = \frac{ab}{4} . \text{ Then }, 3 \odot \left( \frac{1}{5} \odot \frac{1}{2} \right)\] is equal to __________ .
Which of the following is true ?
The law a + b = b + a is called _________________ .
An operation * is defined on the set Z of non-zero integers by \[a * b = \frac{a}{b}\] for all a, b ∈ Z. Then the property satisfied is _______________ .
On Z an operation * is defined by a * b = a2 + b2 for all a, b ∈ Z. The operation * on Z is _______________ .
Consider the binary operation * defined by the following tables on set S = {a, b, c, d}.
| * | a | b | c | d |
| a | a | b | c | d |
| b | b | a | d | c |
| c | c | d | a | b |
| d | d | c | b | a |
Show that the binary operation is commutative and associative. Write down the identities and list the inverse of elements.
Determine whether * is a binary operation on the sets-given below.
a * b – a.|b| on R
On Z, define * by (m * n) = mn + nm : ∀m, n ∈ Z Is * binary on Z?
Fill in the following table so that the binary operation * on A = {a, b, c} is commutative.
| * | a | b | c |
| a | b | ||
| b | c | b | a |
| c | a | c |
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) ∧ C
Let M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}` and let * be the matrix multiplication. Determine whether M is closed under *. If so, examine the commutative and associative properties satisfied by * on M
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
Choose the correct alternative:
A binary operation on a set S is a function from
Choose the correct alternative:
If a * b = `sqrt("a"^2 + "b"^2)` on the real numbers then * is
Let * be a binary operation defined on Q. Find which of the following binary operations are associative
a * b = `"ab"/4` for a, b ∈ Q.
Let * be a binary operation defined on Q. Find which of the following binary operations are associative
a * b = a – b + ab for a, b ∈ Q
Let * be a binary operation defined on Q. Find which of the following binary operations are associative
a * b = ab2 for a, b ∈ Q
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative
a * b = a – b ∀ a, b ∈ Q
The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N 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.
