Advertisements
Advertisements
प्रश्न
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 ?
उत्तर
Commutativity :
\[\text{ Let }a, b \in Q_0 . \text{Then}, \]
\[a o b = \frac{ab}{2}\]
\[ = \frac{ba}{2}\]
\[ = b o a\]
\[\text{Therefore},\]
\[a o b = b o a, \forall a, b \in Q_0\]
Thus, o is commutative on Qo.
Associativity:
\[\text{ Let }a, b, c \in Q_0 . \text{ Then }, \]
\[a o \left( b o c \right) = a o \left( \frac{bc}{2} \right)\]
\[ = \frac{a\left( \frac{bc}{2} \right)}{2}\]
\[ = \frac{abc}{4}\]
\[\left( a o b \right) o c = \left( \frac{ab}{2} \right) o c\]
\[ = \frac{\left( \frac{ab}{2} \right)c}{2}\]
\[ = \frac{abc}{4}\]
\[\text{Therefore},\]
\[a o \left( b o c \right) = \left( a o b \right) o c, \forall a, b, c \in Q_0 \]
Thus, o is associative on Qo.
APPEARS IN
संबंधित प्रश्न
Let A = Q ✕ Q, where Q is the set of all rational numbers, and * be a binary operation defined on A by (a, b) * (c, d) = (ac, b + ad), for all (a, b) (c, d) ∈ A.
Find
(i) the identity element in A
(ii) the invertible element of A.
(iii)and hence write the inverse of elements (5, 3) and (1/2,4)
State whether the following statements are true or false. Justify.
For an arbitrary binary operation * on a set N, a * a = ∀ a a * N.
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.
Determine whether the following operation define a binary operation on the given set or not : '*' on N defined by a * b = a + b - 2 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.
Let * be a binary operation on N given by a * b = LCM (a, b) for all a, b ∈ N. Find 5 * 7.
Check the commutativity and associativity of the following binary operation'*' on Q defined by a * b = ab + 1 for all a, b ∈ Q ?
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 Z defined by a * b = a + b − ab for all a, b ∈ Z ?
The binary operation * is defined by \[a * b = \frac{ab}{7}\] on the set Q of all rational numbers. Show that * is associative.
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 Z defined by
a * b = a + b − 4 for all a, b ∈ Z Show that '*' is both commutative and associative ?
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 :
Find the invertible elements in A ?
Let * be the binary operation on N defined by a * b = HCF of a and b.
Does there exist identity for this binary operation one N ?
Construct the composition table for ×4 on set S = {0, 1, 2, 3}.
Construct the composition table for ×5 on Z5 = {0, 1, 2, 3, 4}.
For the binary operation ×10 on set S = {1, 3, 7, 9}, find the inverse of 3.
For the binary operation ×7 on the set S = {1, 2, 3, 4, 5, 6}, compute 3−1 ×7 4.
Define an associative binary operation on a set.
Mark the correct alternative in the following question:-
For the binary operation * on Z defined by a * b = a + b + 1, the identity element is ________________ .
Which of the following is true ?
The law a + b = b + a is called _________________ .
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 ________________ .
The number of binary operation that can be defined on a set of 2 elements is _________ .
If * is defined on the set R of all real number by *: a * b = `sqrt(a^2 + b^2)` find the identity element if exist in R with respect to *
Let A = {a + `sqrt(5)`b : a, b ∈ Z}. Check whether the usual multiplication is a binary operation on A
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:
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 = a – b + ab 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)2 ∀ a, b ∈ Q
Let A = N x N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Then * is ____________.
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.
Consider the binary operation * on Q defind by a * b = a + 12b + ab for a, b ∈ Q. Find 2 * `1/3`.
