Let W be the set of all continuous fuctions from the set of real numbers R to R. For f, g in W define f+g by (f+g)(x)=f(x)+g(x) and (f*g)=f(x)g(x). Verify that W is a commutative ring. Does W have unity? Why or why not?
To verify if W is a commutative ring, we need to demonstrate that:
1. W is a ring.
2. The addition operation is commutative.
3. The multiplication operation is commutative.
Let's start by showing that W is a ring:
1. W is closed under addition:
For any f and g in W, (f+g)(x) = f(x) + g(x), which is a sum of two real numbers. Since the sum of two real numbers is always a real number, (f+g)(x) is a real number. Hence, f+g is a continuous function, and W is closed under addition.
2. W is closed under multiplication:
For any f and g in W, (f*g)(x) = f(x) * g(x), which is a product of two real numbers. Since the product of two real numbers is always a real number, (f*g)(x) is a real number. Hence, f*g is a continuous function, and W is closed under multiplication.
3. Addition in W is associative:
For any f, g, and h in W, ((f+g)+h)(x) = (f+g)(x) + h(x) = (f(x) + g(x)) + h(x). By associativity of real number addition, this is equal to f(x) + (g(x) + h(x)), which is equal to (f(x) + g(x)) + h(x) = f(x) + (g+h)(x). Therefore, addition in W is associative.
4. Additive identity exists in W:
The additive identity in W is the function defined by the constant mapping f(x) = 0 for all x in R. For any f in W, (f + 0)(x) = f(x) + 0 = f(x) for all x in R. Therefore, the additive identity exists in W.
5. Additive inverse exists in W:
For any f in W, the additive inverse function, denoted as -f, is defined as (-f)(x) = -f(x) for all x in R. It is clear that (-f)(x) + f(x) = 0 for all x in R. Therefore, the additive inverse exists in W.
Now let's check the commutativity of addition and multiplication:
6. Addition is commutative:
For any f and g in W, (f+g)(x) = f(x) + g(x) = g(x) + f(x) = (g+f)(x). Therefore, addition is commutative in W.
7. Multiplication is commutative:
For any f and g in W, (f*g)(x) = f(x) * g(x) = g(x) * f(x) = (g*f)(x). Therefore, multiplication is commutative in W.
Lastly, we need to determine if W has unity (multiplicative identity):
8. W does not have unity:
In order for a ring to have a unity element, there must exist an identity element under multiplication. However, in W, there is no continuous function that acts as the multiplicative identity for all functions in W. This is because the value of a function at a particular point is determined by the output of both the function and the point itself, which cannot be captured by a single continuous function. Therefore, W does not have unity.
In conclusion, we have verified that W is a commutative ring, but it does not have unity.