Let f : R R→Z be the closed binary operation defined
by f (a, b) _ _a + b_. (a) Is f commutative? (b) Is f associative?
(c) Does f have an identity element?
(a) To check if the binary operation is commutative, we need to verify if f(a, b) = f(b, a) for all a and b in the domain.
Let's substitute the values into the definition of f:
f(a, b) = a + b
f(b, a) = b + a
Since addition is commutative in the real numbers, we have a + b = b + a for all real numbers a and b. Therefore, f(a, b) = f(b, a), and the binary operation f is commutative.
(b) To check if the binary operation is associative, we need to verify if f(a, f(b, c)) = f(f(a, b), c) for all a, b, and c in the domain.
Let's substitute the values into the definition of f:
f(a, f(b, c)) = a + (b + c)
f(f(a, b), c) = (a + b) + c
Since addition is associative in the real numbers, we have a + (b + c) = (a + b) + c for all real numbers a, b, and c. Therefore, f(a, f(b, c)) = f(f(a, b), c), and the binary operation f is associative.
(c) To check if the binary operation has an identity element, we need to find an element e such that f(a, e) = a and f(e, a) = a for all a in the domain.
Let's substitute the values into the definition of f:
f(a, e) = a + e
f(e, a) = e + a
For f to have an identity element, we need a + e = a, and e + a = a for all real numbers a. This is only true if e is the additive identity element itself, which is 0 in the real numbers.
Therefore, the binary operation f does not have an identity element.
To determine whether the given operation f is commutative, associative, and if it has an identity element, we need to check certain properties of the operation.
(a) Commutativity:
A binary operation f is commutative if for all elements a and b in the domain, f(a, b) = f(b, a).
To check if f is commutative, we need to compare f(a, b) with f(b, a). In this case, f(a, b) = a + b and f(b, a) = b + a.
Comparing a + b with b + a, we see that a + b = b + a for all real numbers a and b. Therefore, f is commutative.
(b) Associativity:
A binary operation f is associative if for all elements a, b, and c in the domain, f(a, f(b, c)) = f(f(a, b), c).
To check if f is associative, we need to compare f(a, f(b, c)) with f(f(a, b), c). In this case, f(a, b) = a + b.
Comparing f(a, f(b, c)) = f(a, b + c) = a + (b + c) with f(f(a, b), c) = f(a + b, c) = (a + b) + c, we observe that a + (b + c) = (a + b) + c for all real numbers a, b, and c. Therefore, f is associative.
(c) Identity element:
An identity element e of a binary operation f is an element such that for all elements a in the domain, f(a, e) = a = f(e, a).
To check if f has an identity element, we need to find an element e that satisfies f(a, e) = a = f(e, a) for all real numbers a.
Let's assume e is an identity element. Then we have:
f(a, e) = a + e = a and f(e, a) = e + a = a.
From these equations, we can see that e must be 0 since a + 0 = a = 0 + a for all real numbers a.
Therefore, the operation f has an identity element, which is 0.
In summary:
(a) The operation f is commutative.
(b) The operation f is associative.
(c) The operation f has an identity element, which is 0.