Answer to Question #220267 in Discrete Mathematics for Michel

Question

Let U = {l, 2, 3, 4, 5, 6, 7, 8, 9, and 10} be a universal set. Let A, B, C such that A= {l, 3, 4, 8},

B = {2, 3, 4, 5, 9, 10}, and C = {3, 5, 7, 9, 10}. Use bit representations(computer representation) for A, B, and C together with UNION, intersection, difference, and complement to find the bit representation for the following:

(a) AU B

(b) An B n C

(f)) (AU C) n B

(d) (A - B) UC

(e) An (B - (C n B))

(f) A - (B - C)

(g) (AU B) U (C - B)

Answer 1

Solution:

Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, and 10}.

U has 10 elements, so bit representation will contain 10 digits with 0 and 1.

$x_{i}= \begin{cases}1 & \text { if } i \in X \\ 0 & \text { for } i \notin X\end{cases}$

A= {1, 3, 4, 8}, B = {2, 3, 4, 5, 9, 10}, and C = {3, 5, 7, 9, 10}

$(a) A\cup B=\{1,2,3,4,5,8,9,10\}=1111100111 \\(b)A\cap B\cap C=\{3\}=0010000000 \\(c)(A\cup C)\cap B=\{1,3,4,5,7,8,9,10\}\cap \{2, 3, 4, 5, 9, 10\}=\{3,4,5,9,10\}=0011100011 \\(d) (A-B)\cup C=\{1,8\} \cup \{3, 5, 7, 9, 10\}=\{1,3,5,7,8,9,10\}=1010101111$

$(e) A\cap (B - (C \cap B))=\{1, 3, 4, 8\}\cap (\{2, 3, 4, 5, 9, 10\}-\{3,5,9,10\}) \\=\{1, 3, 4, 8\}\cap \{2\}=\phi=0000000000 \\(f) A - (B - C)=\{1, 3, 4, 8\}-\{2,4\}=\{1,3,8\}=1010000100 \\(g) (A\cup B) \cup (C - B)=\{1,2,3,4,5,8,9,10\}\cup\{7\}=\{1,2,3,4,5,7,8,9,10\} \\=1111101111$