Advertisements
Advertisements
प्रश्न
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.
उत्तर
Let X = {0, 1, 2, 3, 4, 5}.
The operation * on X is defined as:
a * b = `{ (a+b, "if a+b < 0"), (a+ b - 6, "if a + b >= 6"):}`
An element e ∈ X is the identity element for the operation *, if a * e = a = e * a ∀ a ∈ X
For a ∈ X we observed that
a * 0 = a + 0 =a [a ∈ X ⇒ a + 0 < 6]
0 * a = 0 + a = a [a ∈ X ⇒ 0 + a < 6]
:. a * 0 =a = 0 * a ∀ a ∈ X
Thus, 0 is the identity element for the given operation *.
An element a ∈ X is invertible if there exists b∈ X such that a * b = 0 = b * a.
ie `{(a+b = 0= b + a, if a + b < 6),(a+6 - 6= 0=b + a - 6, if a+b >= 6):}`
i.e.,
a = −b or b = 6 − a
But, X = {0, 1, 2, 3, 4, 5} and a, b ∈ X. Then, a ≠ −b.
∴b = 6 − a is the inverse of a &mnForE; a ∈ X.
Hence, the inverse of an element a ∈X, a ≠ 0 is 6 − a i.e., a−1 = 6 − a.
APPEARS IN
संबंधित प्रश्न
Let * be a binary operation, on the set of all non-zero real numbers, given by `a** b = (ab)/5` for all a,b∈ R-{0} that 2*(x*5)=10
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.
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?
Consider the binary operations*: R ×R → and o: R × R → R defined as a * b = |a - b| and ao b = a, &mnForE;a, b ∈ R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that &mnForE;a, b, c ∈ R, a*(b o c) = (a* b) o (a * c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.
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).
Number of binary operations on the set {a, b} are
(A) 10
(B) 16
(C) 20
(D) 8
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 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 '*' be a binary operation on N defined by a * b = 1.c.m. (a, b) for all a, b ∈ N
Check the commutativity and associativity of '*' on N.
Check the commutativity and associativity of the following binary operation 'o' on Q defined by \[\text{a o b }= \frac{ab}{2}\] for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = a + ab for all a, b ∈ Q ?
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 Q defined by a * b = ab + 1 for all a, b ∈ Q ?
If the binary operation o is defined by aob = a + b − ab on the set Q − {−1} of all rational numbers other than 1, shown that o is commutative on Q − [1].
Show that the binary operation * on Z defined by a * b = 3a + 7b is not commutative ?
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 Q, the set of all rational numbers a binary operation * is defined by \[a * b = \frac{a + b}{2}\] Show that * is not associative on Q.
Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Find the invertible elements in Z ?
Let A = R0 × R, where R0 denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows (a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R0 × R :
Show that '⊙' is commutative and associative on A ?
Let 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by \[a o b = \frac{ab}{2}, \text{for all a, b} \in Q_0\].
Show that 'o' is both commutative and associate ?
Let R0 denote the set of all non-zero real numbers and let A = R0 × R0. If '*' is a binary operation on A defined by
(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A
Find the identity element in A ?
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.
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.
Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as \[a * b = \begin{cases}a + b & ,\text{ if a + b} < 6 \\ a + b - 6 & , \text{if a + b} \geq 6\end{cases}\]
Show that 0 is the identity for this operation and each element a ≠ 0 of the set is invertible with 6 − a being the inverse of a.
Define a binary operation on a set.
Write the inverse of 5 under multiplication modulo 11 on the set {1, 2, ... ,10}.
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 __________ .
Let * be a binary operation defined on set Q − {1} by the rule a * b = a + b − ab. Then, the identify element for * is ____________ .
For the binary operation * defined on R − {1} by the rule a * b = a + b + ab for all a, b ∈ R − {1}, the inverse of a is ________________ .
Let A = ℝ × ℝ and let * be a binary operation on A defined by (a, b) * (c, d) = (ad + bc, bd) for all (a, b), (c, d) ∈ ℝ × ℝ.
(i) Show that * is commutative on A.
(ii) Show that * is associative on A.
(iii) Find the identity element of * in A.
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 ∧ B
Choose the correct alternative:
Subtraction is not a binary operation in
Choose the correct alternative:
In the set R of real numbers ‘*’ is defined as follows. Which one of the following is not a binary operation on R?
Which of the following is not a binary operation on the indicated set?
a * b = `((a + b))/2` ∀a, b ∈ N is
