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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?
Verified Answer
Let * be the binary operation on the rational number given by a*b=a+b+...
Explanation: For identity e, a+e=e+a=e, a*e = a+e+ae = a => e=0 and e+a = e+a+ea = a => e=0. So e=0 will be identity, for e to be identity, a*e = a ⇒ a+e+ae = a ⇒ e+ae = 0 and e(1+a) = 0 which gives e=0 or a=-1. So, when a = -1, no identity element exist as e can be any value in that case.
Most Upvoted Answer
Let * be the binary operation on the rational number given by a*b=a+b+...
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.
Then W is not closed under vector addition.
So, W is not a subspace of R3.
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Let * be the binary operation on the rational number given by a*b=a+b+...
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,
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,
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