Which of the following are binary operation.justify your answer.
(i) the operation . defined on Q by a.b =a(b-a)
(ii) the operation . defined on [0, pi] by x.y = cosxy
also,for those operation which are binary operations,check whether they are associated and commutative .
clearly (i) is binary op, since rationals are closed under multiplication and addition.
(ii) is not, since x.y might produce a value not in [0,pi]. For any value xy in[pi/2,pi], cos(xy) < 0, so is not in the domain.
I think you can answer the other questions:
is a.(b+c) = a.b + a.c?
is a.b = b.a ?
To determine whether the given operations are binary operations, we need to check if they satisfy two criteria:
1. The operation should be closed: This means that when we apply the operation to any two elements of the given set, the result should also belong to the same set.
2. The operation should be well-defined: This means that the operation should give a unique result for any pair of elements in the set.
Let's analyze each operation:
(i) the operation "." defined on Q by a.b = a(b - a)
To check if this operation is closed, we need to verify if a.b belongs to Q for any a, b belonging to Q. Consider a = 3 and b = 2. Plugging these values into the operation, we get 3.2 = 3(2-3) = -3, which is not in Q. Therefore, the operation is not closed, and it does not satisfy the first criterion. Hence, it is not a binary operation.
(ii) the operation "." defined on [0, pi] by x.y = cos(xy)
To check if this operation is closed, we need to verify if x.y belongs to [0, pi] for any x, y belonging to [0, pi]. Taking any two values, let's say x = 1 and y = 2, we can calculate 1.2 = cos(1*2) = cos(2), which lies between 0 and pi. Therefore, the operation is closed and satisfies the first criterion.
To check associativity, we need to verify if (x.y).z = x.(y.z) holds for all x, y, z in [0, pi]. In this case, we can use the properties of cosine to show that the operation is associative. Since cosine is associative, the operation defined by the product of cosine is also associative.
To check commutativity, we need to verify if x.y = y.x holds for all x, y in [0, pi]. In this case, we can see that cos(xy) = cos(yx) holds, so the operation is commutative.
In conclusion, the operation defined by x.y = cos(xy) is a binary operation, and it is both associative and commutative.