Let X be any finite set with n elements and P(X) the family of subsets of X.
Let F2 be the field of two elements. We define the following binary operations:
+ : P(X) × P(X) → P(X)
- : F2 × P(X) → P(X),

given by:
For any two subsets A and B of X, we have that A + B is the symmetric difference of A and B, that is, the set (A∪B) (A∩B) that has the elements of A or of B, but they are not in A ∩ B.
For any subset A of X, we have 1 · A = A and 0 · A = ∅.

Solve the following items.

a) Show that P(X) is a vector space over F2 with these operations. Show that the family B of unit sets of X is a basis for P(X). What then is the dimension of P(X)?
b) Show that the family W of subsets of X with cardinality even is a subspace of P(X). Find a basis for W and its dimension.
c) Complete the base of W to a base B' of P(X). Give a subspace W' such that P(X) = W ⊕ W'
d) Find the base change matrix from B to B'  and its inverse.

Answer rating: 100% (QA)

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