ejercicios sobre teoría de conjuntos

Teoría de Conjuntos

Bachiller:

Hurtado Valentina

C.I: 23.997.291

Republica Bolivariana de VenezuelaMinisterio del Poder Popular para la Educación SuperiorI.U.P “Santiago Mariño”Escuela de Ing. de Sistemas.Sede Barcelona.

Para empezar se debe tener claro que…

Un conjunto es la reunión de objetos bien definidos y diferenciables entre si, que se encuentran en un momento dado.

A continuación los siguientes conceptos:

• Unión:  Se llama unión de dos conjuntos A y B al conjunto formado por objetos que son elementos de A o de B, es decir:

A u B

• Intersección: Se llama intersección de dos conjuntos A y B al conjunto formado por objetos que son elementos de A y de B, es decir:

A ∩ B

Es el conjunto que contiene a todos los elementos de A que al mismo tiempo están en B.

Utilizaremos las siguientes leyes de Conjuntos:

Propiedades Unión Intersección

Idempotencia A u A= A A ∩ A= A

Conmutativa A u B= B u A A ∩ B= B ∩ A

Asociativa A u (B u C)= (A u B) u C A ∩ (B ∩ C) = (A ∩ B) ∩ C

Distributiva A u (B ∩ C)= (A u B) ∩ (A u C)

A ∩ (B u C) = (A ∩ B) u (A ∩ C)

Complementariedad A u A’ = U A ∩ A’ = Ø

Utilizaremos los conjuntos:

A= (2, 6 , 8, 10, 13, 14, 27)B= (1, 6, 11, 14, 20, 27, 30)C= (4, 6, 8, 11, 17, 19, 22, 27, 35, 40, 41)D= ( A, B, C)

Ejercicios

1.Idempotencia

Formula: A u A= A

A u A= (2, 6 , 8, 10, 13, 14, 27) u (2, 6 , 8, 10, 13, 14, 27)

= (2, 6 , 8, 10, 13, 14, 27)

2. Conmutativa

Formula A u B= B u A

A u B= A + B – A ∩ BA u B= (2, 6 , 8, 10, 13, 14, 27) +(1, 6, 11, 14, 20, 27, 30) – (6, 14, 27)A u B= (1, 2,8, 10, 11, 13, 20, 27, 30)

Esto es igual a:

B u A= B + A – B ∩ AB u A= (1, 6, 11, 14, 20, 27, 30) + (2, 6 , 8, 10, 13, 14, 27) - (6, 14, 27)B u A= (1, 2,8, 10, 11, 13, 20, 27, 30)

3. Asociativa

FormulaA u (B u C)= (A u B) u C

(B u C)= B + C – B ∩ C(B u C)= (1, 6, 11, 14, 20, 27, 30) + (4, 6, 8, 11, 17, 19, 22, 27, 35, 40, 41) – (6, 11, 27)B u C= (1, 6, 11, 14, 20, 27, 30, 4, 8, 17, 19, 22, 35, 40, 41)

A u (B u C)= (2, 6 , 8, 10, 13, 14, 27) + (1, 6, 11, 14, 20, 27, 30, 4, 8, 17, 19, 22, 35, 40, 41) – (6, 8, 14, 27)

A u ( B u C)= (1, 2, 4, 6, 8, 10, 11, 13, 14, 17, 19, 20, 22, 27, 30,35, 40, 41)

4. DistributivaFormula

A u (B ∩ C)= (A u B) ∩ (A u C)

Como A u B es conmutativa(B ∩ C)= (1, 6, 11, 14, 20, 27, 30) ∩ (4, 6, 8, 11, 17, 19, 22, 27, 35, 40, 41)(B ∩ C)= ( 6, 11,27)

A u (B ∩ C)= {A+ ( B ∩ C) } – A ∩ (B ∩ C)A u (B ∩ C)= (2, 6 , 8, 10, 13, 14, 27) + ( 6, 11,27)A u (B ∩ C)= (2, 6, 8, 10, 11, 13, 14, 27)

4.1 Distributiva

A u (B ∩ C)= (A u B) ∩ (A u C) si la formula es cumplida es

distributiva.

A u B= (2, 6 , 8, 10, 13, 14, 27) + (1, 6, 11, 14, 20, 27, 30) - (6, 14, 27) A u B= (1, 2,8, 10, 11, 13, 20, 27, 30)

A u C= A + C- A ∩ CA u C= (2, 6 , 8, 10, 13, 14, 27) + (4, 6, 8, 11, 17, 19, 22, 27, 35, 40, 41) – (6, 8, 27)A u C= (2, 4, 6, 8, 10, 11, 13, 14, 17, 19, 22, 27, 35, 40, 41)

(A u B) ∩ (A u C)= (1, 2,8, 10, 11, 13, 20, 27, 30) ∩ (2, 4, 6, 8, 10, 11, 13, 14, 17, 19, 22, 27, 35, 40, 41)(A u B) ∩ (A u C)= (2, 8, 10, 11, 13, 27)

5.Complementariedad Formula A ∩ A’

= Ø

D= AD= (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27)A= (2, 6 , 8, 10, 13, 14, 27)

A’= (1, 3, 4, 5, 7, 9, 11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26)

A u A’= A + A’

5.1 ComplementariedadSi A u A’ = Ø

el conjunto es de

complementariedad.

A u A’= A + A’A u A’= (2, 6 , 8, 10, 13, 14, 27) + (1, 3, 4, 5, 7, 9, 11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26)

A u A’= (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27)