Advertisements
Advertisements
प्रश्न
Which of the following is true ?
पर्याय
* defined by \[a * b = \frac{a + b}{2}\] is a binary operation on Z .
* defined by \[a * b = \frac{a + b}{2}\] is a binary operation on Q .
all binary commutative operations are associative.
subtraction is a binary operation on N.
Advertisements
उत्तर
Let us check each option one by one.
* defined by \[a * b = \frac{a + b}{2}\] is a binary operation on Z .
\[\text{If }a = 1 \text{ and } b = 2, \]
\[a * b = \frac{a + b}{2}\]
\[ = \frac{1 + 2}{2}\]
\[ = \frac{3}{2} \not\in Z\]
Hence, it is false.
* defined by \[a * b = \frac{a + b}{2}\] is a binary operation on Q .
\[a * b = \frac{a + b}{2} \in Q, \forall a, b \in Q\]
\[\text { For example: Let a } = \frac{3}{2}, b = \frac{5}{6} \in Q\]
\[a * b = \frac{\frac{3}{2} + \frac{5}{6}}{2}\]
\[ = \frac{9 + 5}{12}\]
\[ = \frac{14}{12}\]
\[ = \frac{7}{6} \in Q\]
Hence, it is true.
all binary commutative operations are associative.
Commutativity :-
\[\text{ Let a, b } \in N . \text{ Then }, \]
\[a * b = 2^{ab} \]
\[ = 2^{ba} \]
\[ = b * a\]
\[\text{ Therefore },\]
\[a * b = b * a, \forall a, b \in N\]
Thus, * is commutative on N.
Associativity :-
\[\text{ Let a, b, c } \in N . \text{ Then }, \]
\[a * \left( b * c \right) = a * \left( 2^{bc} \right)\]
\[ = 2^{a^* 2^{bc}} \]
\[\left( a * b \right) * c = \left( 2^{ab} \right) * c\]
\[ = 2^{ab^* 2^c} \]
\[\text{ Therefore },\]
\[a * \left( b * c \right) \neq \left( a * b \right) * c\]
Thus, * is not associative on N.
Therefore, all binary commutative operations are not associative.
Hence, it is false.
subtraction is a binary operation on N.
Subtraction is not a binary operation on N because subtraction of any two natural numbers is not always a natural number.
For example : 2 and 4 are natural numbers.
2 − 4 = −2 which is not a natural number.
Hence, it is false.
APPEARS IN
संबंधित प्रश्न
For each binary operation * defined below, determine whether * is commutative or associative.
On Q, define a * b = ab + 1
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.
State whether the following statements are true or false. Justify.
If * is a commutative binary operation on N, then a * (b * c) = (c * b) * a
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 for all a, b ∈ N.
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 the following operation define a binary operation on the given set or not :
\[' * ' \text{on Q defined by } a * b = \frac{a - 1}{b + 1} \text{for all a, b} \in Q .\]
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 = ab
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.
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 ?
Check the commutativity and associativity of the following binary operations '*'. on Q defined by a * b = a − b 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 = (a − b)2 for all a, b ∈ Q ?
Check the commutativity and associativity of the following binary operation '*' on N, defined by a * b = ab for all a, b ∈ N ?
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.
The binary operation * is defined by \[a * b = \frac{ab}{7}\] on the set Q of all rational numbers. Show that * is associative.
Find the identity element in the set of all rational numbers except −1 with respect to *defined by a * b = a + b + ab.
Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z Show that '*' is both commutative and associative ?
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 commutative binary operation 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.
Let +6 (addition modulo 6) be a binary operation on S = {0, 1, 2, 3, 4, 5}. Write the value of \[2 +_6 4^{- 1} +_6 3^{- 1} .\]
If the binary operation * on the set Z of integers is defined by a * b = a + 3b2, find the value of 2 * 4.
If a * b = a2 + b2, then the value of (4 * 5) * 3 is _____________ .
Let * be a binary operation on Q+ defined by \[a * b = \frac{ab}{100} \text{ for all a, b } \in Q^+\] The inverse of 0.1 is _________________ .
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 _______________ .
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.
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 * 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
Let * be a binary operation on Q, defined by a * b `= (3"ab")/5` 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.
Let * be the binary operation on N given by a * b = HCF (a, b) where, a, b ∈ N. Find the value of 22 * 4.
