Advertisements
Advertisements
प्रश्न
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 ?
उत्तर
\[\text{ Let }a \in Q - \left\{ - 1 \right\} \text{ and } b \in Q - \left\{ - 1 \right\} \text{be the inverse of a} . \text{ Then }, \]
\[a * b = e = b * a\]
\[ \Rightarrow a * b = e \text{ and }b * a = e\]
\[ \Rightarrow a + b + ab = 0 \text{ and }b + a + ba = 0\]
\[ \Rightarrow b\left( 1 + a \right) = - a \in Q - \left\{ - 1 \right\}\]
\[ \Rightarrow b = \frac{- a}{1 + a} \in Q - \left\{ - 1 \right\} \left[ \because a \neq - 1 \right]\]
\[\text{Thus},\frac{- a}{1 + a} \text{is the inverse of a} \in Q - \left\{ - 1 \right\} . \]
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)
Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this.
On Z+, define * by a * b = a
Let A = N × N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d)
Show that * is commutative and associative. Find the identity element for * on A, if any.
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?
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 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 = ab2
Here, Z+ denotes the set of all non-negative integers.
Is * defined on the set {1, 2, 3, 4, 5} by a * b = LCM of a and b a binary operation? Justify your answer.
Determine which of the following binary operations are associative and which are commutative : * on Q defined by \[a * b = \frac{a + b}{2} \text{ for all a, b } \in Q\] ?
Check the commutativity and associativity of the following binary operations '*'. on N defined by a * b = 2ab for all a, b ∈ N ?
Check the commutativity and associativity of the following binary operation '*' on Q defined by \[a * b = \frac{ab}{4}\] 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].
The binary operation * is defined by \[a * b = \frac{ab}{7}\] on the set Q of all rational numbers. Show that * is associative.
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 Show that '*' is both commutative and associative ?
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}.
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.
Write the composition table for the binary operation multiplication modulo 10 (×10) on the set S = {2, 4, 6, 8}.
Which of the following is true ?
Let * be a binary operation on R defined by a * b = ab + 1. Then, * is _________________ .
An operation * is defined on the set Z of non-zero integers by \[a * b = \frac{a}{b}\] for all a, b ∈ Z. Then the property satisfied 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 ________________ .
On the set Q+ of all positive rational numbers a binary operation * is defined by \[a * b = \frac{ab}{2} \text{ for all, a, b }\in Q^+\]. The inverse of 8 is _________ .
Let * be a binary operation defined on Q+ by the rule
\[a * b = \frac{ab}{3} \text{ for all a, b } \in Q^+\] The inverse of 4 * 6 is ___________ .
Determine whether * is a binary operation on the sets-given below.
a * b – a.|b| on R
Define an operation * on Q as follows: a * b = `(("a" + "b")/2)`; a, b ∈ Q. Examine the existence of identity and the existence of inverse for the operation * on Q.
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) v C
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
Choose the correct alternative:
Subtraction is not a binary operation in
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 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
The identity element for the binary operation * defined on Q – {0} as a * b = `"ab"/2 AA "a, b" in "Q" - {0}` is ____________.
If * is a binary operation on the set of integers I defined by a * b = 3a + 4b - 2, then find the value of 4 * 5.
Which of the following is not a binary operation on the indicated set?
a * b = `((a + b))/2` ∀a, b ∈ N is
