contestada

Let A = {a, b, c}, B = {b, c, d}, and C = {b, c, e}. (a) Find A ∪ (B ∩ C), (A ∪ B) ∩ C, and (A ∪ B) ∩ (A ∪ C). (Enter your answer in set-roster notation.) A ∪ (B ∩ C) = (A ∪ B) ∩ C = (A ∪ B) ∩ (A ∪ C) = Which of these sets are equal? A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (A ∪ B) ∩ C = (A ∪ B) ∩ (A ∪ C) A ∪ (B ∩ C) = (A ∪ B) ∩ C (b) Find A ∩ (B ∪ C), (A ∩ B) ∪ C, and (A ∩ B) ∪ (A ∩ C). (Enter your answer in set-roster notation.) A ∩ (B ∪ C) = (A ∩ B) ∪ C = (A ∩ B) ∪ (A ∩ C) = Which of these sets are equal? A ∩ (B ∪ C) = (A ∩ B) ∪ C (A ∩ B) ∪ C = (A ∩ B) ∪ (A ∩ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (c) Find (A − B) − C and A − (B − C). (Enter your answer in set-roster notation.) (A − B) − C = A − (B − C) = Are these sets equal? Yes No

Respuesta :

MrRoyal

Answer:

(a)

[tex]A\ u\ (B\ n\ C) = \{a,b,c\}[/tex]

[tex](A\ u\ B)\ n\ C = \{b,c\}[/tex]

[tex](A\ u\ B)\ n\ (A\ u\ C) = \{b,c\}[/tex]

[tex](A\ u\ B)\ n\ C = (A\ u\ B)\ n\ (A\ u\ C)[/tex]

(b)

[tex]A\ n\ (B\ u\ C) = \{b,c\}[/tex]

[tex](A\ n\ B)\ u\ C = \{b,c,e\}[/tex]

[tex](A\ n\ B)\ u\ (A\ n\ C) = \{b,c\}[/tex]

[tex]A\ n\ (B\ u\ C) = (A\ n\ B)\ u\ (A\ n\ C)[/tex]

(c)

[tex](A - B) - C = \{a\}[/tex]

[tex]A - (B - C) = \{a,b,c\}[/tex]

They are not equal

Step-by-step explanation:

Given

[tex]A= \{a,b,c\}[/tex]

[tex]B =\{b,c,d\}[/tex]

[tex]C = \{b,c,e\}[/tex]

Solving (a):

[tex]A\ u\ (B\ n\ C)[/tex]

[tex](A\ u\ B)\ n\ C[/tex]

[tex](A\ u\ B)\ n\ (A\ u\ C)[/tex]

[tex]A\ u\ (B\ n\ C)[/tex]

B n C means common elements between B and C;

So:

[tex]B\ n\ C = \{b,c,d\}\ n\ \{b,c,e\}[/tex]

[tex]B\ n\ C = \{b,c\}[/tex]

So:

[tex]A\ u\ (B\ n\ C) = \{a,b,c\}\ u\ \{b,c\}[/tex]

u means union (without repetition)

So:

[tex]A\ u\ (B\ n\ C) = \{a,b,c\}[/tex]

Using the illustrations of u and n, we have:

[tex](A\ u\ B)\ n\ C[/tex]

[tex](A\ u\ B)\ n\ C = (\{a,b,c\}\ u\ \{b,c,d\})\ n\ C[/tex]

Solve the bracket

[tex](A\ u\ B)\ n\ C = (\{a,b,c,d\})\ n\ C[/tex]

Substitute the value of set C

[tex](A\ u\ B)\ n\ C = \{a,b,c,d\}\ n\ \{b,c,e\}[/tex]

Apply intersection rule

[tex](A\ u\ B)\ n\ C = \{b,c\}[/tex]

[tex](A\ u\ B)\ n\ (A\ u\ C)[/tex]

In above:

[tex]A\ u\ B = \{a,b,c,d\}[/tex]

Solving A u C, we have:

[tex]A\ u\ C = \{a,b,c\}\ u\ \{b,c,e\}[/tex]

Apply union rule

[tex]A\ u\ C = \{b,c\}[/tex]

So:

[tex](A\ u\ B)\ n\ (A\ u\ C) = \{a,b,c,d\}\ n\ \{b,c\}[/tex]

[tex](A\ u\ B)\ n\ (A\ u\ C) = \{b,c\}[/tex]

The equal sets

We have:

[tex]A\ u\ (B\ n\ C) = \{a,b,c\}[/tex]

[tex](A\ u\ B)\ n\ C = \{b,c\}[/tex]

[tex](A\ u\ B)\ n\ (A\ u\ C) = \{b,c\}[/tex]

So, the equal sets are:

[tex](A\ u\ B)\ n\ C[/tex] and [tex](A\ u\ B)\ n\ (A\ u\ C)[/tex]

They both equal to [tex]\{b,c\}[/tex]

So:

[tex](A\ u\ B)\ n\ C = (A\ u\ B)\ n\ (A\ u\ C)[/tex]

Solving (b):

[tex]A\ n\ (B\ u\ C)[/tex]

[tex](A\ n\ B)\ u\ C[/tex]

[tex](A\ n\ B)\ u\ (A\ n\ C)[/tex]

So, we have:

[tex]A\ n\ (B\ u\ C) = \{a,b,c\}\ n\ (\{b,c,d\}\ u\ \{b,c,e\})[/tex]

Solve the bracket

[tex]A\ n\ (B\ u\ C) = \{a,b,c\}\ n\ (\{b,c,d,e\})[/tex]

Apply intersection rule

[tex]A\ n\ (B\ u\ C) = \{b,c\}[/tex]

[tex](A\ n\ B)\ u\ C = (\{a,b,c\}\ n\ \{b,c,d\})\ u\ \{b,c,e\}[/tex]

Solve the bracket

[tex](A\ n\ B)\ u\ C = \{b,c\}\ u\ \{b,c,e\}[/tex]

Apply union rule

[tex](A\ n\ B)\ u\ C = \{b,c,e\}[/tex]

[tex](A\ n\ B)\ u\ (A\ n\ C) = (\{a,b,c\}\ n\ \{b,c,d\})\ u\ (\{a,b,c\}\ n\ \{b,c,e\})[/tex]

Solve each bracket

[tex](A\ n\ B)\ u\ (A\ n\ C) = \{b,c\}\ u\ \{b,c\}[/tex]

Apply union rule

[tex](A\ n\ B)\ u\ (A\ n\ C) = \{b,c\}[/tex]

The equal set

We have:

[tex]A\ n\ (B\ u\ C) = \{b,c\}[/tex]

[tex](A\ n\ B)\ u\ C = \{b,c,e\}[/tex]

[tex](A\ n\ B)\ u\ (A\ n\ C) = \{b,c\}[/tex]

So, the equal sets are:

[tex]A\ n\ (B\ u\ C)[/tex] and [tex](A\ n\ B)\ u\ (A\ n\ C)[/tex]

They both equal to [tex]\{b,c\}[/tex]

So:

[tex]A\ n\ (B\ u\ C) = (A\ n\ B)\ u\ (A\ n\ C)[/tex]

Solving (c):

[tex](A - B) - C[/tex]

[tex]A - (B - C)[/tex]

This illustrates difference.

[tex]A - B[/tex] returns the elements in A and not B

Using that illustration, we have:

[tex](A - B) - C = (\{a,b,c\} - \{b,c,d\}) - \{b,c,e\}[/tex]

Solve the bracket

[tex](A - B) - C = \{a\} - \{b,c,e\}[/tex]

[tex](A - B) - C = \{a\}[/tex]

Similarly:

[tex]A - (B - C) = \{a,b,c\} - (\{b,c,d\} - \{b,c,e\})[/tex]

[tex]A - (B - C) = \{a,b,c\} - \{d\}[/tex]

[tex]A - (B - C) = \{a,b,c\}[/tex]

They are not equal