Consider the following two subspaces of RR U f R rightarrow

Consider the following two subspaces of R^R: U = {f: R rightarrow R: f(x) = f(x+2)} and V= {(f:R rightarrow R: f(x) = f(x+3)}. In other words, these are the subspaces of functions of period 2 and 3, respectively. Is there a period p such that U + V subsector equal to {f: R rightarrow R: f(x) = f(x + p)}? Determine p and prove your result, OR, give a counterexample.

Solution

The correct answer will be LCM(2,3) = 6

Reason:

For the function U (f(x) = f(x+2))

f(x) = f(x+2(3)) = f(x+6), since 2 is the period for the function

For the function V((f(x) = f(x+3))

f(x) =f(x+3(2)) = f(x+6), since 3 is the period for the function

Hence, U+V will combinely have the period equal to 6