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प्रश्न
Choose the correct alternative:
A binary operation on a set S is a function from
पर्याय
S → S
(S × S) → S
S → (S × S)
(S × S) → (S × S)
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उत्तर
(S × S) → S
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संबंधित प्रश्न
For each binary operation * defined below, determine whether * is commutative or associative.
On Q, define a * b = ab + 1
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.
Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by A * B = A ∩ B &mnForE; A, B in P(X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P(X) with respect to the operation*.
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.
Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = ab2 for all a, b ∈ Q ?
Let S be the set of all rational numbers except 1 and * be defined on S by a * b = a + b \[-\] ab, for all a, b \[\in\] S:
Prove that * is a binary operation on S ?
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 ?
For the binary operation ×7 on the set S = {1, 2, 3, 4, 5, 6}, compute 3−1 ×7 4.
For the binary operation multiplication modulo 5 (×5) defined on the set S = {1, 2, 3, 4}. Write the value of \[\left( 3 \times_5 4^{- 1} \right)^{- 1}.\]
Consider the binary operation * defined on Q − {1} by the rule
a * b = a + b − ab for all a, b ∈ Q − {1}
The identity element in Q − {1} is _______________ .
If * is defined on the set R of all real numbers by *: a*b = `sqrt(a^2 + b^2 ) `, find the identity elements, if it exists in R with respect to * .
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 existence of identity, existence of inverse properties for the operation * on M
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
Let * be a binary operation defined on Q. Find which of the following binary operations are associative
a * b = a – b for a, b ∈ Q
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative
a * b = a2 + b2 ∀ a, b ∈ Q
If the binary operation * is defined on the set Q + of all positive rational numbers by a * b = `" ab"/4. "Then" 3 "*" (1/5 "*" 1/2)` is equal to ____________.
Determine which of the following binary operation on the Set N are associate and commutaive both.
