Advertisements
Advertisements
प्रश्न
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
उत्तर
Let a, b ∈ A
i.e. a ≠ ±1 , b ≠ 1
Now a * b = a + b – ab
If a + b – ab = 1
⇒ a + b – ab – 1 = 0
i.e. a(1 – b) – 1(1 – b) = 0
(a – 1)(1 – b) = 0 ⇒ a = 1, b = 1
But a ≠ 1 , b ≠ 1
So (a – 1)(1 – 6) ≠ 1
i.e. a * b ∈ A.
So * is a binary on A.
To verify the commutative property:
Let a, b ∈ A
i.e. a ≠ 1, b ≠ 1
Now a * b = a + b – ab
And b * a = b + a – ba
So a * b = b * a
⇒ * is commutative on A.
To verify the associative property:
Let a, b, c ∈ A
i.e. a, b, c ≠ 1
To prove the associative property we have to prove that
a * (b * c) = (a * b) * c
L.H.S: b * c = b + c – bc = D .......(say)
So a * (b * c) = a * D = a + D – aD
= a + (b + c – bc) – a(b + c – bc)
= a + b + c – bc – ab – ac + abc
= a + b + c – ab – bc – ac + abc .......(1)
R.H.S: (a * b) = a + b – ab = K ......(say)
So (a * b) * c = K * c = K + c – Kc
= (a + b – ab) + c – (a + b – ab)c
= a + b – ab + c – ac – bc + abc
= a + b + c – ab – bc – ac + abc ......(2)
APPEARS IN
संबंधित प्रश्न
Let * be a binary operation, on the set of all non-zero real numbers, given by `a** b = (ab)/5` for all a,b∈ R-{0} that 2*(x*5)=10
For each binary operation * defined below, determine whether * is commutative or associative.
On Z, define a * b = a − b
Determine whether the following operation define a binary operation on the given set or not :
\[' +_6 ' \text{on S} = \left\{ 0, 1, 2, 3, 4, 5 \right\} \text{defined by}\]
\[a +_6 b = \begin{cases}a + b & ,\text{ if a} + b < 6 \\ a + b - 6 & , \text{if a} + b \geq 6\end{cases}\]
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.
Determine which of the following binary operations are associative and which are commutative : * on Q defined by \[a * b = \frac{a + b}{2} \text{ for all a, b } \in Q\] ?
On the set Q of all ration numbers if a binary operation * is defined by \[a * b = \frac{ab}{5}\] , prove that * is associative on Q.
If the binary operation * on the set Z is defined by a * b = a + b −5, the find the identity element with respect to *.
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.
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.
Define identity element for a binary operation defined on a set.
Write the composition table for the binary operation multiplication modulo 10 (×10) on the set S = {2, 4, 6, 8}.
If a * b = a2 + b2, then the value of (4 * 5) * 3 is _____________ .
If G is the set of all matrices of the form
\[\begin{bmatrix}x & x \\ x & x\end{bmatrix}, \text{where x } \in R - \left\{ 0 \right\}\] then the identity element with respect to the multiplication of matrices as binary operation, is ______________ .
On Z an operation * is defined by a * b = a2 + b2 for all a, b ∈ Z. The operation * on Z is _______________ .
Let * be binary operation defined on R by a * b = 1 + ab, ∀ a, b ∈ R. Then the operation * is ______.
If the binary operation * is defined on the set Q + of all positive rational numbers by a * b = `" ab"/4. "Then" 3 "*" (1/5 "*" 1/2)` is equal to ____________.
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?
