Boolean Function with ^ and or

Question

Please provide feedback on my answer to this question. Question: Prove that not every boolean function is equal to a boolean function constructed by only using ^ and or. Answer: True, Suppose that a function f(m,n)=m exclusive n using only ^ and or. Such that:m=1,n=1 Then:m exclusive n = 0 Since: m ^ n = 1 and m or n = 1 Thus: m exclusive n = 0 is not equal to a boolean function constructed by ^ and or such that:p^q = 1 and p or q = 1. Sorry I cant type the symbol for and , or and exclusive..

Answer 1

This seems to make sense. Basically you're showing that XOR cannot be constructed from just ANDs and ORs, because $1 \oplus 1 = 0$, yet $1\cdot 1 = 1 + 1 = 1$. If you want to make this more rigorous, you can prove this using induction on the complexity of Boolean formulas built from ANDs and ORs. You want to show that for every Boolean function on two variables $f$ composed only of ANDs and ORs, $f(1, 1) = 1$.