Answer to Question #132257 in Discrete Mathematics for Dinu Wijesinghe

Question

1. 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
for A, B, and C together with UNION,
INTER, DIFF, and COMP 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

$A=\{1,3,4,8\}$

$1\ 0\ 1\ 1\ 0\ 0\ 0\ 1\ 0\ 0$

$B=\{2,3,4,5,9,10\}$

$0\ 1\ 1\ 1\ 1\ 0\ 0\ 0\ 1\ 1$

$C=\{3,5,7,9,10\}$

$0\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 1$

Table for the Bit Operators OR,AND, and XOR

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c :c} x & y & x\lor y & x\land y & x\oplus y \\ \hline 0 & 0 & 0 & 0 & 0 \\ \hdashline 0 & 1 & 1 & 0 & 1 \\ \hdashline 1 & 0 & 1 & 0 & 1 \\ \hdashline 1 & 1 & 1 & 1 & 0 \\\end{array}$$

a) $A\cup B$

$1\ 0\ 1\ 1\ 0\ 0\ 0\ 1\ 0\ 0$

$0\ 1\ 1\ 1\ 1\ 0\ 0\ 0\ 1\ 1$

$1\ 1\ 1\ 1\ 1\ 0\ 0\ 1\ 1\ 1$ bitwise $OR$

$A\cup B=\{1,2,3,4,5,8,9,10\}$

b) $A\cap B\cap C$

$1\ 0\ 1\ 1\ 0\ 0\ 0\ 1\ 0\ 0$

$0\ 1\ 1\ 1\ 1\ 0\ 0\ 0\ 1\ 1$

$0\ 0\ 1\ 1\ 0\ 0\ 0\ 0\ 0\ 0$ bitwise $AND$

$0\ 0\ 1\ 1\ 0\ 0\ 0\ 0\ 0\ 0$

$0\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 1$

$0\ 0\ 1\ 0\ 0\ 0\ 0\ 0\ 0\ 0$ bitwise $AND$

$A\cap B\cap C=\{3\}$

c) $(A\cup C)\cap B$

$1\ 0\ 1\ 1\ 0\ 0\ 0\ 1\ 0\ 0$

$0\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 1$

$1\ 0\ 1\ 1\ 1\ 0\ 1\ 1\ 1\ 1$ bitwise $OR$

$1\ 0\ 1\ 1\ 1\ 0\ 1\ 1\ 1\ 1$

$0\ 1\ 1\ 1\ 1\ 0\ 0\ 0\ 1\ 1$

$0\ 0\ 1\ 1\ 1\ 0\ 0\ 0\ 1\ 1$ bitwise $AND$

$(A\cup C)\cap B=\{3,4,5,9,10\}$

d) $(A-B)\cup C$

$1\ 0\ 1\ 1\ 0\ 0\ 0\ 1\ 0\ 0$

$0\ 1\ 1\ 1\ 1\ 0\ 0\ 0\ 1\ 1$

$1\ 0\ 0\ 0\ 0\ 0\ 0\ 1\ 0\ 0$ bitwise $A-B$

$1\ 0\ 0\ 0\ 0\ 0\ 0\ 1\ 0\ 0$

$0\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 1$

$1\ 0\ 1\ 0\ 1\ 0\ 1\ 1\ 1\ 1$ bitwise $OR$

$(A-B)\cup C=\{1,3,5,7,8,9,10\}$

e) $A\cap (B-(C\cap B))$

$0\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 1$

$0\ 1\ 1\ 1\ 1\ 0\ 0\ 0\ 1\ 1$

$0\ 0\ 1\ 0\ 1\ 0\ 0\ 0\ 1\ 1$ bitwise $AND$

$0\ 1\ 1\ 1\ 1\ 0\ 0\ 0\ 1\ 1$

$0\ 0\ 1\ 0\ 1\ 0\ 0\ 0\ 1\ 1$

$0\ 1\ 0\ 1\ 0\ 0\ 0\ 0\ 0\ 0$ bitwise $B-(C\cap B)$

$1\ 0\ 1\ 1\ 0\ 0\ 0\ 1\ 0\ 0$

$0\ 1\ 0\ 1\ 0\ 0\ 0\ 0\ 0\ 0$

$0\ 0\ 0\ 1\ 0\ 0\ 0\ 0\ 0\ 0$ bitwise $AND$

$A\cap (B-(C\cap B))=\{4\}$

f) $A-(B-C)$

$0\ 1\ 1\ 1\ 1\ 0\ 0\ 0\ 1\ 1$

$0\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 1$

$0\ 1\ 0\ 1\ 0\ 0\ 0\ 0\ 0\ 0$ bitwise $B-C$

$1\ 0\ 1\ 1\ 0\ 0\ 0\ 1\ 0\ 0$

$0\ 1\ 0\ 1\ 0\ 0\ 0\ 0\ 0\ 0$

$1\ 0\ 1\ 0\ 0\ 0\ 0\ 1\ 0\ 0$ bitwise $A-(B-C)$

$A-(B-C)=\{1,3,8\}$

g) $(A\cup B)\cup(C-B)$

$1\ 0\ 1\ 1\ 0\ 0\ 0\ 1\ 0\ 0$

$0\ 1\ 1\ 1\ 1\ 0\ 0\ 0\ 1\ 1$

$1\ 1\ 1\ 1\ 1\ 0\ 0\ 1\ 1\ 1$ bitwise $OR$

$0\ 0\ 1\ 0\ 1\ 0\ 1\ 0\ 1\ 1$

$0\ 1\ 1\ 1\ 1\ 0\ 0\ 0\ 1\ 1$

$0\ 0\ 0\ 0\ 0\ 0\ 1\ 0\ 0\ 0$ bitwise $C-B$

$1\ 1\ 1\ 1\ 1\ 0\ 0\ 1\ 1\ 1$

$0\ 0\ 0\ 0\ 0\ 0\ 1\ 0\ 0\ 0$

$1\ 1\ 1\ 1\ 1\ 0\ 1\ 1\ 1\ 1$ bitwise $OR$

$(A\cup B)\cup(C-B)=\{1,2,3,4,5,7,8,9,10\}$