# characteristic functions of subsets satisfy the properties

Prove that the characteristic functions of subsets satisfy the foowing properties:

(a) f_(A intersection B) (x) = f_A (x) f_B (x) for all x,

(b) f_(A union B) (x) = f_A (x) + f_B (x) - f_A (x) f_B (x) for all x,

(c) f_(A symmetric difference B) (x) = f_A (x) + f_B (x) - 2f_A (x) f_B (x)

for all x.

The complete problem is in the attached file.

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#### Solution Preview

The solution is in the attached file.

Descrete Structures (I)

By:- Thokchom Sarojkumar Sinha

Prove that the characteristic functions of subsets satisfy the following

properties :

Solution:- (a) If

then

if and only if

if and only if

if and only if

if and only if

Therefore

for

Again, suppose

if and only if

...

#### Solution Summary

It describes the characteristic functions of subsets satisfy the properties of

(a) f_(A intersection B) = f_A f_B

(b) f_(A union B) = f_A + f_B - f_A f_B

(c) f _(A symmetric difference B) = f_A + f_B - 2f_A f_B

The solution is given in detail.