Advertisements
Advertisements
प्रश्न
Let S be the set of all real numbers except −1 and let '*' be an operation defined by a * b = a + b + ab for all a, b ∈ S. Determine whether '*' is a binary operation on S. If yes, check its commutativity and associativity. Also, solve the equation (2 * x) * 3 = 7.
Advertisements
उत्तर
Checking for binary operation:
\[\text{Let }a, b \in S . \text{Then}, \]
\[a, b \in \text{R and a} \neq - 1, b \neq - 1\]
\[a * b = a + b + ab\]
\[\text{We need to prove thata} + b + ab \in S . \left[ \text{For this we have to prove thata} + b + ab \in \text{ R and a } + b + ab \neq - 1 \right]\]
\[\text{Since a, b} \in R, a + b + ab \in R, \text{let us assume thata} + b + ab = - 1 . \]
\[a + b + ab + 1 = 0\]
\[a + ab + b + 1 = 0\]
\[a\left( 1 + b \right) + 1\left( 1 + b \right) = 0\]
\[\left( a + 1 \right)\left( b + 1 \right) = 0\]
\[a = - 1, b = - 1 \left[ \text{which is false} \right]\]
\[\text{Hence},a + b + ab \neq - 1\]
\[\text{Therefore},\]
\[a + b + ab \in S\]
Thus, * is a binary operation on S.
Commutativity:
\[\text{Let }a, b \in S . \text{Then}, \]
\[a * b = a + b + ab\]
\[ = b + a + ba\]
\[ = b * a \]
\[\text{Therefore},\]
\[a * b = b * a, \forall a, b \in S\]
Thus, * is commutative on N.
Associativity :
\[\text{Let a}, b, c \in S\]
\[a * \left( b * c \right) = a * \left( b + c + bc \right)\]
\[ = a + b + c + bc + a\left( b + c + bc \right)\]
\[ = a + b + c + bc + ab + ac + abc\]
\[\left( a * b \right) * c = \left( a + b + ab \right) * c\]
\[ = a + b + ab + c + \left( a + b + ab \right)c\]
\[ = a + b + ab + c + ac + bc + abc\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in S\]
Thus, * is associative on S.
Now,
\[\text{Given}:\hspace{0.167em}\left( 2 * x \right) * 3 = 7\]
\[ \Rightarrow \left( 2 + x + 2x \right) * 3 = 7\]
\[ \Rightarrow \left( 2 + 3x \right) * 3 = 7\]
\[ \Rightarrow 2 + 3x + 3 + \left( 2 + 3x \right)3 = 7\]
\[ \Rightarrow 5 + 3x + 6 + 9x = 7\]
\[ \Rightarrow 12x + 11 = 7\]
\[ \Rightarrow 12x = - 4\]
\[ \Rightarrow x = \frac{- 4}{12}\]
\[ \Rightarrow x = \frac{- 1}{3}\]
APPEARS IN
संबंधित प्रश्न
LetA= R × 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 identity element for * on A. Also find the inverse of every element (a, b) ε A.
For each binary operation * defined below, determine whether * is commutative or associative.
On Z+, define a * b = 2ab
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 N given by a * b = L.C.M. of a and b. Find
(i) 5 * 7, 20 * 16
(ii) Is * commutative?
(iii) Is * associative?
(iv) Find the identity of * in N
(v) Which elements of N are invertible for the operation *?
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?
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 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+ define * by a * b = |a − b|
Here, Z+ denotes the set of all non-negative integers.
Find the total number of binary operations on {a, b}.
Determine which of the following binary operation is associative and which is commutative : * on N defined by a * b = 1 for all a, b ∈ N ?
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 Q defined by a ⊙ b = a2 + b2 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 Z defined by a * b = a + b − ab for all a, b ∈ Z ?
On Q, the set of all rational numbers, * is defined by \[a * b = \frac{a - b}{2}\] , shown that * is no associative ?
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 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 ?
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 Q − {−1} defined by a * b = a + b + ab for all a, b ∈ Q − {−1} Show that '*' is both commutative and associative on Q − {−1}.
On R − {1}, a binary operation * is defined by a * b = a + b − ab. Prove that * is commutative and associative. Find the identity element for * on R − {1}. Also, prove that every element of R − {1} is invertible.
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
Show that '*' is both commutative and associative on A ?
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 invertible element in A ?
Construct the composition table for ×5 on Z5 = {0, 1, 2, 3, 4}.
Write the total number of binary operations on a set consisting of two elements.
If a * b denote the bigger among a and b and if a ⋅ b = (a * b) + 3, then 4.7 = __________ .
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 ________________ .
Q+ is the set of all positive rational numbers with the binary operation * defined by \[a * b = \frac{ab}{2}\] for all a, b ∈ Q+. The inverse of an element a ∈ Q+ 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 _________________ .
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
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 commutative and associative properties satisfied by * on A
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` ∀ a, b ∈ Q ~ {0} is ______.
A binary operation on a set has always the identity element.
Let * be a binary operation on Q, defined by a * b `= (3"ab")/5` is ____________.
The binary operation * defined on N by a * b = a + b + ab for all a, b ∈ N is ____________.
Which of the following is not a binary operation on the indicated set?
A binary operation A × A → is said to be associative if:-
