1. DISCRETE MATHEMATICS
(SMA3023)
ASSIGNMENT 2
Lecturer’s Name: En. Shahrizal bin Samsuddin
Group : A
Member’s Name:
Name Matric ID
NurFaralinaBintiAsrab Ali D20101037415
Noor AzurahBt Abdul Razak D20101037502
NurWahidahBtSami’on D20101037525

2. Exercise 2.2 p.61. Question 14
If p => q is false, can you determine the truth value of (~p) v (p <=> q).
Explain your answer.
Answer:
Yes can determine the truth value.
If p => q is false,
Then, by using the truth table we know that
p => q
True True True
True False False
False True True
False True False
p is true and q is false.
When p is true,
Hence, (~p) is false.
And by using the truth table we know that
p <=> q
True False False
p <=> q is false.
So, when we use the truth table we know that
(~p) v (p <=> q)
False False False
(~p) v (p <=> q) is false

3. Exercise 2.3 p.68. Question 30
Determine if the following ia a valid argument. Explain your conclusion.
Prove: If x is an irrational number then 1-x is also an irrational number.
Proof: suppose 1-x is irrational, then we can write 1-x as with a,b,€ Z. Now we have 1- = x
and x = a rational number. This is contradiction. Hence, if x is irrational, so is 1-x
Answers:
Direct method
Assume,
p: if x is an irrational number
q: then 1-x is also an irrational number
Assume that,
1- x = , Q’
x=1-
This show that x also an irrational number
Hence the statement is valid

4. Exercise 2.4 p.73. Question 8
Let P(n): 13
+ 23
+ 33
+……+ n3
=
a) Use P(k) to show P(k+1)
b) P(n) is true for all n ≥1
Answers:
LHS of P(k+1): 13
+ 23
+ 33
+……+ (k+1)3
=
a) LHS of P(k+1): 13
+ 23
+ 33
+……+ (k+1)3
= + (k+1)3
= +(k+1)3
= 1 + (k+1)3
= ( +1 + (k+1)3
=RHS of P(k+1)
b) No it is not true because the value from LHS ≠ RHS when we substitute n ≥1

5. Exercise 2.4 p.73. Question 10
Prove 1+2n
< 3n
for n ≥ 2
Answers:
Basis Step : n = 2
P(2): 1+22
< 32
is true
Induction step
1+2k
+1 < 3k
+ 1
21
+ 2k
< 3k
+ 30
< 2k
+2k
< 2(k+1)
RHS of P(k+1)