# DIFERENCIA Propiedades: A−A = Ø 1.X (A−B)

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```DIFERENCIA
• A−A = &Oslash;
1.X (A−B)
2. XA &quot; −XA definici&oacute;n de diferencia
4. A−A = &Oslash; por la definici&oacute;n de diferencia
2) A−(A&quot;B)= A−B
• X [A−(A&quot;B)]XEA &quot;−X (A&quot;B) por la definici&oacute;n de intersecci&oacute;n
• XA &quot; −(XA &quot; XB) por la definici&oacute;n de diferencia
• XA &quot; (−XA &quot; &eth; −XB) de morgan
• (XA &quot; − XA) &quot; X&quot; −XB) contradicci&oacute;n
• F&quot; X (A−B) definici&oacute;n de neutro
• A− (A&quot; B) = A−B definici&oacute;n de diferencia
3) (A−B)
A = (A−B)
1.
X, (X
A&quot; X
B) &quot; X
A por definici&oacute;n de diferencia
2.(X
A&quot; X
B) &quot;(X
A&quot; X
A) distributiva
3.(X
A&quot; X
B) &quot; X
A
4.(X
A&quot; X
B) de la intersecci&oacute;n
5.(A−B) definici&oacute;n de diferencia
4) (A−B)
1
B =A
B
1.
X, (X
A&quot; X
B) &quot; X
B definici&oacute;n de diferencia y d uni&oacute;n
2.(X
A&quot; X
B) &quot; (X
B&quot; X
B) distributiva
3. (X
A&quot; X
B) &quot; V neutro
4.(X
A&quot; X
B) definici&oacute;n de intersecci&oacute;n
5.A
B
5) A−B = (A
B)−B
• De (A
B)−B
1.X, X
[(A
B) −B]
2. X
[(A
B)−B] definici&oacute;n de diferencia
3. X
(A
B) &quot; X
B definici&oacute;n de uni&oacute;n
4.[ X
A &quot;X
B) &quot; [ X
B&quot;X
2
5. X
(A−B) definici&oacute;n de diferencia
LUEGO (A
B)−B
A−B
b) De (A−B)
1.X, X
(A−B)
2. X
A&quot;X
B definici&oacute;n de diferencia
3.[ X
A &quot;X
B) &quot; [ X
B&quot;X
B ] dilema
4.(A
B)−B
A−B de condicional
LUEGO A−B
(A
B)−B tanto son iguales
6) (A
B) −B = &quot;
1.
X, X
(A
B) &quot; X
B definici&oacute;n de diferencia
2. (X
A&quot; X
B) &quot; X
B definici&oacute;n de intersecci&oacute;n
3. (X
A&quot; X
B) &quot; (X
B&quot; X
B) distributiva
4.(X
A&quot; X
3
5.&quot; neutro
• B&quot; (A−B) = &Oslash;
1.XB &quot; (A−B)
2.XB &quot; X (A−B) definici&oacute;n de intersecci&oacute;n
3.XB &quot; (XA &quot; −XB) definici&oacute;n de diferencia
4.(XB &quot; &eth; XB) &quot; XA distributiva
8) A−(B
C) = (A−B)
(A−C)
1.
X, X
A&quot; X
(B
C) definici&oacute;n de diferencia
2.X
A&quot;(X
B&quot; X
C) definici&oacute;n de uni&oacute;n
3.(X
A&quot;X
B) &quot; (X
A&quot; X
C) distributiva
4.(A−B)
(A−C)
9) A−(B
C) = (A−B)
(A−C)
1.
X, X
A&quot; X
(B
C) definici&oacute;n de diferencia
2. X
A&quot; (X
4
B&quot; X
C) definici&oacute;n de intersecci&oacute;n
3.(X
A&quot; X
B) &quot;(X
A&quot;X
C) distributiva
4.A−B)
(A−C)
10) [ [(A
B) − (A
C)]
A
(B−C)
Por probarse dos inclusiones: (A
B) − (A
C)
A
(B−C) &quot; A
(B−C)
(A
B)−(A
C)
Probemos que: [ [(A
B) − (A
C)]
A
(B−C)
•
•
•
x
[(A
B) − (A
C)]
x
(A
B) &quot; x
(A
C)
x
(A
B) &quot; [x
A&quot; x
C)
5
•
•
•
•
x
(A
B) &quot; [x
A'&quot; x
C')
[x
(A
B) &quot; x
A'] &quot; [ x
(A
B) &quot; x
C')
[x
A&quot;x
B] &quot; x
A'] &quot; [x
(A
B) &quot; x
C')
[x
B &quot; (x
A&quot;x
A'] &quot; [x
(A
B) &quot; x
C')
F
•
•
•
•
•
x
(A
B) &quot; x
C')
x
A &quot; [x
B&quot;x
C')
x
A &quot; [x
(B−C)]
x
[A
(B−C)]
6
Por 1 y 11 [ (A
B) − (A
C)]
A
(B−C)
Ahora probemos que: A
(B−C)
(A
B) − (A
C)
•
•
x
[(A
(B−C)] ..................... (hip)
x
A&quot;
(B−C)
•x
A &quot; (x
B &quot;x
C)
• [x
A&quot;x
B] &quot; x
C
•x
(A
B) &quot; x
C
Aplicar la tautologia: F &quot; P = P en particular para F= x
A&quot;x
A
• F &quot; [x
(A
B) &quot;
C']
7
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