Let X be a finite non empty set and V = P(X) = {S | S ⊆ X} the set of all subsets of X. For S, T ∈ V , we define
S + T = (S ∪ T) − (S ∩ T).
Suppose X={x1, x2,...,xn} with distinct xi. Prove that B=({x1}, {x2},..., {xn}) is a basis of V.
Let X be a finite non empty set and V = P(X) = {S | S ⊆ X} the set of all subsets of X. For S, T ∈ V , we define
S + T = (S ∪ T) − (S ∩ T).
Suppose X={x1, x2,...,xn} with distinct xi. Prove that B=({x1}, {x2},..., {xn}) is a basis of V.
A basis for a vector space has two properties: the vectors in the basis span the space and they are independent. It is easy to see how to write any "vector" as a "linear combination" of the x_n and it should be just as easy to see that they are independent.
(At first I missed the "in F2" which is only in your title, not in the body of the question, and wondered about scalar multiplication. Of course, the only members of F2 are 0 and 1. 0 times any vector is 0 and 1 times any vector is the vector itself.)