Advertisements
Advertisements
प्रश्न
Let A be any set containing more than one element. Let '*' be a binary operation on A defined by a * b = b for all a, b ∈ A Is '*' commutative or associative on A ?
उत्तर
\[\text{Let a}, b \in A . \text{Then}, \]
\[a * b = b \]
\[b * a = a\]
\[\text{Therefore},\]
\[a * b \neq b * a\]
Thus, * is not commutative on A.
Associativity:
\[\text{Let } a, b, c \in A . \text{Then}, \]
\[a * \left( b * c \right) = a * c\]
\[ = c\]
\[\left( a * b \right) * c = b * c\]
\[ = c\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in A\]
Thus, * is associative on A.
APPEARS IN
संबंधित प्रश्न
For each binary operation * defined below, determine whether * is commutative or associative.
On Z+, define a * b = 2ab
Consider the binary operation ∨ on the set {1, 2, 3, 4, 5} defined by a ∨b = min {a, b}. Write the operation table of the operation∨.
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 |
Let*′ be the binary operation on the set {1, 2, 3, 4, 5} defined by a *′ b = H.C.F. of a and b. Is the operation *′ same as the operation * defined in Exercise 4 above? Justify your answer.
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.
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 the following operation define a binary operation on the given set or not : '⊙' on N defined by a ⊙ b= ab + ba for all a, b ∈ N
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 Z+, define * by a * b = a
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 = a + 4b2
Here, Z+ denotes the set of all non-negative integers.
Check the commutativity and associativity of the following binary operation '*' on R defined by a * b = a + b − 7 for all a, b ∈ R ?
Check the commutativity and associativity of the following binary operation '*' on Z defined by a * b = a − b for all a, b ∈ Z ?
Check the commutativity and associativity of the following binary operation '*' on Q defined by \[a * b = \frac{ab}{4}\] for all a, b ∈ Q ?
Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative ?
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.
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.
If the binary operation * on the set Z is defined by a * b = a + b −5, the find the identity element with respect to *.
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 Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Find the identity element in Q − {−1} ?
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.
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 the set Z of all integers a binary operation * is defined by a * b = a + b + 2 for all a, b ∈ Z. Write the inverse of 4.
If a * b = a2 + b2, then the value of (4 * 5) * 3 is _____________ .
Which of the following is true ?
A binary operation * on Z defined by a * b = 3a + b for all a, b ∈ Z, 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 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
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 * 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 = ab2 for a, b ∈ Q
Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.
Let * be the binary operation on N given by a * b = HCF (a, b) where, a, b ∈ N. Find the value of 22 * 4.
Consider the binary operation * on Q defind by a * b = a + 12b + ab for a, b ∈ Q. Find 2 * `1/3`.
Subtraction and division are not binary operation on.
