ABOUT THE CHARACTERISTIC FUNCTION OF A SET Prof. Mihály Bencze, Department of Mathematics, University of Braşov, Romania Prof. Florentin Smarandache, Chair of Department of Math & Sciences, University of New Mexico, 200 College Road, Gallup, NM 87301, USA, E-mail: [email protected]

Abstract: In this paper we give a method, based on the characteristic function of a set, to solve some difficult problems of set theory found in undergraduate studies. Definition: Let’s consider A ⊂ E ≠ ∅ (a universal set), then f A : E → {0, 1} , ⎧1, if x ∈ A where the function f A ( x) = ⎨ is called the characteristic function of the set ⎩0, if x ∉ A A. Theorem 1: Let’s consider A, B ⊂ E . In this case f A = fB if and only if A = B . Proof.

⎧1, if x ∈ A = B f A ( x) = ⎨ ⎩0, if x ∉ A = B

= f B ( x)

Reciprocally: For any x ∈ A , fA (x) = 1 , but f A = fB , therefore fB (x) = 1 , namely x ∈ B from where A ⊂ B . The same way we prove that B ⊂ A , namely A = B . Theorem 2: f A% = 1 − f A , A% = CE A . Prof. ⎧⎪1, if x ∈ A% f A% ( x) = ⎨ ⎪⎩0, if x ∉ A%

⎧1, if x ∉ A ⎧1 − 0, if x ∉ A ⎧0, =⎨ =⎨ = 1− ⎨ ⎩0, if x ∈ A ⎩1 − 1, if x ∈ A ⎩1,

Theorem 3: fA ∩ B = fA ∗ fB .

1

if x ∉ A = 1 − f A ( x) if x ∈ A

Proof.

⎧1, if x ∈ A ∩ B f A∩ B ( x ) = ⎨ ⎩0, if x ∉ A ∩ B

⎧1, ⎪ ⎧1, if x ∈ A and x ∈ B ⎪0, =⎨ =⎨ ⎩0, if x ∉ A or x ∉ B ⎪0, ⎪⎩0,

⎛ ⎧1 if x ∈ A ⎞ = ⎜⎨ ⎟ ⎝ ⎩0 if x ∉ A ⎠

if if if if

x ∈ A, x ∈ A, x ∉ A, x ∉ A,

x∈B x∉ B = x∈ B x∉B

⎛ ⎧1 if x ∈ B ⎞ ⎜⎨ ⎟ = f A ( x) f B ( x) . ∉ x B 0 if ⎩ ⎝ ⎠

The theorem can be generalized by induction: n

Theorem 4: f n

I Ak

= ∏ f Ak k =1

k =1

Consequence. For any n ∈ N ∗ , f Mn = f M . Proof. In the previous theorem we chose A1 = A2 = ... = An = M . Theorem 5: f A∪ B = f A + f B − f A f B . Proof. f A∪ B = f A∪ B = f A∩ B = 1 − f A∩ B = 1 − f A f B = 1 − (1 − f A )(1 − f B ) = f A + f B − f A f B

It can be generalized by induction: n

Theorem 6: f n

U Ak

= ∑ (−1) k −1

k =1

k =1

n

∑

1≤i1