Define characteristic function and prove that:

Question

Define characteristic function and prove that:

(i) f A'(x) = 1 – f A(x)

(ii) f AUB = f A(x) + f B(x) – f A(x) .f B(x)

Answer 1

The characteristic function of a set A is the function that takes the value 1 if x is an element of A and the value 0 if x is not an element of A.

i) Suppose that x does not belong to A. Hence, x belongs to A' which means that the LHS of i) is equal to 1 and the RHS of i) is equal to 1-0=1. Now, if x belongs to A, then x does not belong to the set A' which means that the LHS of i) is 0 and the RHS of i) is equal to 1-1=0. So the equality is proven.

ii) If x belongs to the union of A and B then either x belongs to A or to B or to both of them. In the 1st scenario we have that the LHS of ii) is equal to 1 and the RHS of ii) is equal to 1+0-0=1. In the 2nd scenario we have that the LHS of ii) is equal to 1 and the RHS of ii) is equal to 0+1-0=1. In the 3rd scenario we have that the LHS of ii) is equal to 1 and the RHS of ii) is equal to 1+1-1=1.

The last case is when x does not belong to the union of A and B which means that x does not belong to A and x does not belong to B. But then we have that the LHS of ii) is equal to 0 and the RHS of ii) is equal to 0+0-0=0.

The equality is proven.