Advertisements
Advertisements
Question
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 commutative as well as associative ?
Advertisements
Solution
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 S.
Associativity:
\[\text{Let} a, b, c \in S . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( b + c - bc \right)\]
\[ = a + b + c - bc - a\left( b + c - bc \right)\]
\[ = a + b + c - bc - ab - ac + abc\]
\[\left( a * b \right) * c = \left( a + b - ab \right) * c\]
\[ = a + b - ab + c - \left( a + b - ab \right)c\]
\[ = a + b + c - ab - ac - bc + abc\]
\[\text{Therefore}, \]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in S\]
Thus , * is associative on S.
So, * is commutative as well as associative.
APPEARS IN
RELATED QUESTIONS
Determine whether or not 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 = ab
For each binary operation * defined below, determine whether * is commutative or associative.
On Q, define a * b = ab + 1
For each binary operation * defined below, determine whether * is commutative or associative.
On Z+, define a * b = ab
Find which of the operations given above has identity.
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.
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
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 : 'O' on Z defined by a O b = ab for all a, b ∈ Z.
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.
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.
Let S = {a, b, c}. Find the total number of binary operations on S.
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 = a2 + b2 for all a, b ∈ Q ?
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 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.
On the set Z of integers, if the binary operation * is defined by a * b = a + b + 2, then find the identity element.
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 'o' be a binary operation on the set Q0 of all non-zero rational numbers defined by \[a o b = \frac{ab}{2}, \text{ for all a, b } \in Q_0\] :
Find the identity element in Q0.
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.
For the binary operation multiplication modulo 10 (×10) defined on the set S = {1, 3, 7, 9}, write the inverse of 3.
If the binary operation * on the set Z of integers is defined by a * b = a + 3b2, find the value of 2 * 4.
The binary operation * is defined by a * b = a2 + b2 + ab + 1, then (2 * 3) * 2 is equal to ______________ .
On Z an operation * is defined by a * b = a2 + b2 for all a, b ∈ Z. The operation * on Z 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 ___________ .
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.
Define an operation * on Q as follows: a * b = `(("a" + "b")/2)`; a, b ∈ Q. Examine the closure, commutative and associate properties satisfied by * 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
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 commutative and associative properties satisfied by * on M
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
Choose the correct alternative:
A binary operation on a set S is a function from
Let N be the set of natural numbers. Then, the binary operation * in N defined as a * b = a + b, ∀ a, b ∈ N has identity element.
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative
a * b = a + ab ∀ a, b ∈ Q
The binary operation * defined on set R, given by a * b `= "a+b"/2` for all a, b ∈ R is ____________.
The identity element for the binary operation * defined on Q – {0} as a * b = `"ab"/2 AA "a, b" in "Q" - {0}` 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.
