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## Homework Statement

Decide which of the following are subrings of the ring of all functions from the closed interval [0,1] to R (the reals)

a) The set of all functions f(x) such that f(q) = 0 for all q in Q (the rationals) & q in [0, 1]

b) The set of all polynomial functions

c) The set of all functions which have only a finite number of zeros, together with the zero function

d) The set of all functions which have an infinite number of zeros.

e) The set of all functions f such that lim {x -> 1+} f(x) = 0

f) The set of all rational linear combinations of the functions sin(nx) and cos(mx) where m and n are non-negative integers

## Homework Equations

## The Attempt at a Solution

The first one is pretty straight forward to show that it is a subring. But unless I'm mistaken, the sets mentioned from b to f aren't even subsets let alone subgroups or subrings since they can be defined on a larger domain than [0,1]. Am I correct? Or am I reading it wrong?