Discrete Mathematics Assignment Solutions

Running head: DISCRETE MATHEMATICS
DISCRETE MATHEMATICS
Name of the Student
Name of the University
Author Note

DISCRETE MATHEMATICS1
Question 1:
a) (175773)10 to ()5
This can be obtained by dividing the number by 5 and then taking out the remainders in
descending order.
Divisor Dividend Remainder
5 175773 3
5 35154 4
5 7030 0
5 1406 1
5 281 1
5 11 1
5 2 2
0
(175773)10 = (21111043)5
b) (22.5602)10 can be converted to base 8 by dividing the number by 8 and then taking the
remainders.
Divisor Dividend Remainder
8 22 6
2
product remainder whole

DISCRETE MATHEMATICS2
number
0.5602*8 4.4816 0.4816 4
0.4816*8 3.8528 0.8528 3
0.8528*8 6.8224 0.8224 6
0.8528*8 6.5792 0.5792 6
0.5792*8 4.6336 0.6336 4
(22.5602)10 = (26.43664)8
ii) (FBCF9)16 + (3AFD7)16
a)
In the hexadecimal addition if the sum is more than F then carry is added to the previous
addition bit.
FBCF9
3AFD7
= 136CD0
b) (5432)8 – (574)8 = (4636)8
Octal subtraction is like the subtraction in any other number system. Only difference is in the
borrow part, in decimal number system a group of 1010 is borrowed and in octal number
system a group of 810 is borrowed.
c) (110010101)2 X (1100)2
Binary multiplication rules are
0X0 = 0, 1X0 = 0, 0X1 = 0 and 1X1 = 1.

DISCRETE MATHEMATICS3
This follows the same multiplication structure of decimal multiplication where each digit of
one number is multiplied with every digit of other number in corresponding one bit shifted
lines and finally all those lines are added.
Hence, (110010101)2 X (1100)2 = (1001011111100)2
Question 2:
i. A compound statement is a tautology if the statement is true irrespective of the truth of
individual statements. Compound statement is a contradiction if the statement is always false
irrespective of the values of individual statements. A compound statement is satisfiable if at
least one truth statement for one or more truth values of the individuals.
In the statement (~𝑎∨~𝑏) ˅ (𝑎∧𝑏) the individual statements are a, b, (~𝑎∨~𝑏) and (𝑎∧𝑏). In the
truth table 1 is considered truth and 0 is considered false.
a b (~𝑎∨~𝑏) (𝑎∧𝑏) (~𝑎∨~𝑏) ˅ (𝑎∧𝑏)
0 0 1 0 1
0 1 1 0 1
1 0 1 0 1
1 1 0 1 1
Hence, the compound statement is always truth irrespective of the truth of individual
statement and hence the compound statement is a tautology.
ii. (𝑝∨~𝑞)∧(𝑝∧𝑞)≡(𝑝∧𝑞)
Truth table:
p q (𝑝∨~𝑞) (𝑝∧𝑞) (𝑝∨~𝑞)∧(𝑝∧𝑞)
0 0 1 0 0