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 .
To determine which of the given operations are binary operations, we need to verify if the operations satisfy the definition of a binary operation.
A binary operation is a rule or function that takes two elements from a set and combines them to produce a unique result. In other words, for a binary operation to be defined on a set, it must take two elements from that set and produce a third element that is also in the set.
Let's evaluate each operation to see if they meet this criteria:
(i) The operation . defined on Q by a.b = a(b - a).
To check if this operation is a binary operation, we need to ensure that when we take two elements from Q (the set of rational numbers) and apply the operation on them, the resulting value also belongs to Q.
Let's take two rational numbers, a and b, and apply the operation: a.b = a(b - a)
For any two rational numbers a and b, (b - a) will also be a rational number, as subtraction of two rational numbers gives a rational number.
Then, multiplying a by (b - a) will also give us a rational number.
Therefore, the operation . defined on Q is a binary operation.
(ii) The operation . defined on [0, pi] by x.y = cos(xy).
To determine if this operation is a binary operation, we need to verify that when we take two elements from the interval [0, pi] and apply the operation on them, the resulting value also falls within [0, pi].
When we multiply two numbers in [0, pi] and then take the cosine of the product, the resulting angle will still be between 0 and pi because the cosine function is bounded by -1 and 1.
Therefore, the operation . defined on [0, pi] is a binary operation.
To check if the binary operations are associative, we need to verify if (a.b).c = a.(b.c) holds for all a, b, c within the given set.
For the operation . defined on Q by a.b = a(b - a):
(a.b).c = a(b-c)
a.(b.c) = a((b-a)-c) = a(b-a-c)
Since (a.b).c and a.(b.c) are equal, the operation is associative.
For the operation . defined on [0, pi] by x.y = cos(xy):
(x.y).z = cos((xy).z) = cos(x(yz))
x.(y.z) = cos(x(y.z))
Since (x.y).z and x.(y.z) are equal, the operation is associative.
To check if the binary operations are commutative, we need to verify if a.b = b.a holds for all a, b within the given set.
For the operation . defined on Q by a.b = a(b - a):
a.b = a(b - a) = ab - a^2
b.a = b(a - b) = ba - b^2
The operations ab - a^2 and ba - b^2 are not equivalent, so the operation is not commutative.
For the operation . defined on [0, pi] by x.y = cos(xy):
x.y = cos(xy)
y.x = cos(yx)
Since cos(xy) and cos(yx) are not necessarily equal, the operation is not commutative.
In summary:
(i) The operation . defined on Q by a.b = a(b - a) is a binary operation. It is associative but not commutative.
(ii) The operation . defined on [0, pi] by x.y = cos(xy) is a binary operation. It is associative but not commutative.