Let * be the binary operation on the rational number given by a*b=a+b+ab. Which of the following property does not exist for the group?

a)
Closure property

b)
Identity property

c)
Symmetric property

d)
Associative property

Correct answer is option 'B'. Can you explain this answer?

Question

Let * be the binary operation on the rational number given by a*b=a+b+ab. Which of the following property does not exist for the group?

a)
Closure property
b)
Identity property
c)
Symmetric property
d)
Associative property

Correct answer is option 'B'. Can you explain this answer?

Answer

For the set W= {(x, 1,z) ε R³}.
Then W is not closed under vector addition.
So, W is not a subspace of R³.

Answer

For the set W= {(x, 1,z) ε R³}.
Then W is not closed under vector addition.
So, W is not a subspace of R³.

Answer

Let ∗ be the binary operation on the rational numbers given by a∗b=a+b+ab. We are asked to identify the property that does not exist for the group.

The operation does not satisfy the identity property since there is no element such that e∗a=a∗e=a for all a in the group.

Answer

Closure Property:

The closure property states that for any two elements a and b in a set, the result of the binary operation on a and b will also be in the set. In other words, if we perform the operation * on any two rational numbers, the result will still be a rational number.

In this case, let's take two rational numbers, say a = p/q and b = r/s, where p, q, r, and s are integers and q and s are nonzero.

Now, let's perform the operation * on a and b:
a * b = (p/q) * (r/s) * (p/q) (r/s)
= (p * r * p * r) / (q * s * q * s)
= (p^2 * r^2) / (q^2 * s^2)

Since p^2, r^2, q^2, and s^2 are all integers, and q^2 and s^2 are nonzero, the result of the operation * is still a rational number. Therefore, the closure property holds for the given binary operation.

Identity Property:

The identity property states that there exists an identity element in the set such that when this element is operated with any other element, the result is the other element itself. In other words, there exists an element e in the set such that for any element a in the set, a * e = e * a = a.

Let's find the identity element for the given binary operation.
Let e be the identity element. Then, for any rational number a, we have:
a * e = a
(a * e) * (a * e) = a * a
a * (e * a) * e = a * a
(a * a) * (e * e) = a * a

From the above equation, we can see that (e * e) will be equal to the identity element. However, there is no rational number whose square is equal to itself. Therefore, there does not exist an identity element for the given binary operation.

Symmetric Property:

The symmetric property states that for any two elements a and b in a set, if a is related to b, then b is also related to a. In other words, if a * b = b * a, then b * a = a * b.

In the given binary operation, let's consider two rational numbers a = p/q and b = r/s.

a * b = (p/q) * (r/s) * (p/q) * (r/s)
= (p^2 * r^2) / (q^2 * s^2)

b * a = (r/s) * (p/q) * (r/s) * (p/q)
= (r^2 * p^2) / (s^2 * q^2)

We can see that (p^2 * r^2) / (q^2 * s^2) is equal to (r^2 * p^2) / (s^2 * q^2), which means that a * b = b * a. Therefore, the symmetric property holds for the given binary operation.

Associative Property:

The associative property states that for any three elements a,

Answer

Closure Property
The closure property states that for any two elements a and b in the set, the result of the binary operation will also be in the set. In other words, if a and b are rational numbers, then the product of a and b will also be a rational number.

In the given binary operation *, a*b = a b ab. Since a, b, and ab are all rational numbers, the product of a and b will also be a rational number. Therefore, the closure property is satisfied for this binary operation.

Identity Property
The identity property states that there exists an element called the identity element, such that when it is combined with any other element, the result is the other element itself. In other words, for any element a in the set, there exists an element e, such that a * e = e * a = a.

In the given binary operation *, let's try to find the identity element. Let e be the identity element. Then, for any rational number a, we have a * e = a e ae.

If we substitute a = 0, we get 0 * e = 0 e 0e.

But in order for the identity property to hold, we need 0 * e = 0. This would imply that 0e = 0, which is only true if e = 1. However, if we substitute a = 1, we get 1 * e = 1 e e. This implies that e = e^2, which is only true if e = 1 or e = 0.

Therefore, there is no unique identity element in this binary operation and hence, the identity property does not exist.

Symmetric Property
The symmetric property is not applicable to groups. It is a property of relations, not operations. Therefore, we cannot say that the symmetric property does not exist for this group since it is not a relevant property.

Associative Property
The associative property states that for any three elements a, b, and c in the set, the order in which the binary operation is performed does not matter. In other words, (a * b) * c = a * (b * c).

Let's test the associative property for the given binary operation. Consider rational numbers a, b, and c. We have:

(a * b) * c = (a b ab) * c = (a b ab) c (a b ab)c = (a b ab)(c^2) = a b ab c^2 = a (b c) ac^2 = a * (b * c)

Therefore, the associative property is satisfied for this binary operation.

Conclusion
Based on the analysis above, we can conclude that the binary operation * on the rational numbers does not satisfy the identity property, which means the correct answer is option B.

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