Discrete Mathematics Answers

2. Let P(x) and Q(x) be the statements x3 < 50 and x+2 < 5 respectively. Find the truth values of following quantifications, where the domain consists of all real numbers.
(i) ∃x P(x)
(ii) ∀x P(x)
(iii) ∀x Q(x)
(iv) ∃x Q(x)
(v) ∀x (¬P(x))
(vi) ∃x (¬Q(x))
(vii) ∃x> 0 (P(x))
(viii) ∀x> 0 (¬Q(x))
(ix) ∀x (P(x) ∨ Q(x))
(x) ∃x (P(x) ∧¬Q(x))

Discuss how you can efficiently use Graph Theory to construct a route planning of a project for a vacation trip from Colombo to Trincomalee by considering most of the practical situations (such as millage of the vehicle, etc.) as much as you can. Essentially consider the two fold,
- Routes with shortest distance(Quick route travelling by own vehicle)
- Route with the lowest cost

Which of the following pairs of statements are logically equivalent? I.
Statement 1: It is not true that either I watched 'Crashlanding on You' or you watched 'Money Heist'.
Statement 2: I did not watch 'Crashlanding on You' and you watched 'Money Heist'.

II.Statement 1: Joe tells you that he is an engineer and he studied in Mapua.
Statement 2: If Joe is lying, then Joe is not an engineer or he did not study in Mapua.

Let n be a natural number. Use mathematical induction to prove that
4
n−1 > n2
for all n ≥ 3.

(a) Let P(x) be the statement x
2 ≥ x.
(i) What are truth values of the propositions P(1) , P(-1) , P(0) and P( 1
2
)?
(ii) What is the truth value of the proposition ∀xP(x), where the domain consists of all
real numbers?
(iii) What is the truth value of the proposition ∀xP(x), where the domain consists of all
integers?
(b) Let Q(x,y,z) be the statement x
2 + y
2 = z
2
. What are the truth values of
(i) ∀(x,y,z) Q(x,y,z)
(ii) ∃(x,y,z) Q(x,y,z) Where the domain consists of all real numbers.

What is the De Morgan’s law for quantifiers?
(b) Write the negation of following statements.
(i) ∃x (x
2 + 2 < 1)
(ii) ∀x (x - 2 ≥ 3)
(iii) ∀x (x
2 ≥ 0 and x + 2 < 1)
(iv) ∃x (x
2 + 2 < 0 or x - 5 ≥ 0)
(v) ∀x (x - 2 ≥ 3 and x
3 + 5 ≤ 2)
(vi) ∀x (x
2 = 2)
(vii) There is a student in our class who likes chemistry.
(viii) Every fox is cunning.

Using direct method, prove the following.
(i) If n is an even integer then -n is even.
(ii) If n is an even integer then 3n + 5 is odd.
(iii) If n is an odd integer then n
2 + 3n is even.
(iv) If m is an even integer and n is an odd integer then m2
- 2n is even.
(v) If m and n are even integers then mn + r is an odd integer. Where r is an odd
integer.
(vi) The sum of any two odd integers is even.
(vii) Let x and y be positive real numbers. If x ≤ y then √
x ≤
√y.
(viii) For any integers a,b and c if a divides b and b divides c then a divides c.
(Hint - if a divides b then we can write b = ka for some integer k)

Combinations:
(a) A class contains 10 students with 6 men and 4 women. Find the number of ways to:
i. Select a 4-member committee from the students
ii. Select a 4-member committee with 2 men and 2 women
iii. Elect a president, vice president, and secretary
(b) A box contains 8 blue socks and 6 red socks. Find the number of ways two socks can be drawn
from the box if:
i. They can be any color.
ii. They must be the same color.

There are four train routes between A and B, and three train routes between B and C. Find
the number of ways that a person can travel by train:
i. from A to C by way of B;
ii. roundtrip from A to C by way of B;
iii. roundtrip from A to C by way of B without using a train lines more than once.

1. Discuss two examples on binary trees both quantitatively and qualitatively.