Discrete Mathematics Answers

(b) (i) Let A = {2,4,6} and R = {(2,2), (2,4), (2,6), (4,4), (6,6)} be a relation on A.
Find R−1
.
(ii) Let A = {6,8,10,15} and B = {2,3,4}. Define a relation R from A to B by
(x,y) ∈ R iff x-y is divisible by 2. Find R−1

(a) Let A = {0,1,2}. R = {(0,0), (0,1), (0,2), (1,1), (1,2), (2,2)} and
S = {(0,0), (1,1), (2,2)} be two relations on A.
(i) Show that R is a partial order relation.
(ii) Is R a total order relation?
(iii) Show that S is an equivalence relation.
1

(a) Let A = {2,4,6} and B = {x,y,z}. State each of the following are relations from A into B.
(i) R1 = {(2,x), (y,4), (6,z)}
(ii) R2 = {(4,y), (y,4)}
(iii) R3 = {(2,x), (4,y), (6,z)}
(iv) R4 = {(4,y), (6,x), (4,x)}
(v) R5 = {(x,2), (4,z), (2,z), (6,y)}

(b) Let A be the set {1,2,3,4,5,6}.Which orders pairs are in the following relations on A.
(i) R1 = {(a,b) : a divides b}
(ii) R2 = {(a,b) : a ≤ b}
(iii) R3 = {(a,b) : a
2 = b}
(iv) R4 = {(a,b) : a + b ≤ 5}
(v) R5 = {(a,b) : a = 2 and a + 2b ≤ 10}

Q1 Draw any balance rooted tree having nodes from a to i. (2+2+2+2)
Find the level of each vertex in the drawn tree.
Find the height of this tree.
Which vertices are descendants of node g.

Q3 X people are chosen from a volley ball team (Take a value of X by yourself, possibly that number must be close to a number of volley ball team members). (2+2+2)
a) How many ways are there to choose Y people to take them to ground(Take value of y by yourself less than x)

b) How many ways are there to assign Z positions by selecting players from X people.(Take Z value by yourself and previous X value.)


c) Of the X people T are women. How many ways are there to choose W players to take them to the field if at least 1 of these players must be a women (take help from example 15)


Hint:
First fill all the values of x,y,z,t and w then your question will be in a mathematical form then you can easily solve them.

Q2 Draw any three graphs (Take help from book, but DO NOT copy paste any graph from examples or exercise. Your graphs must be random and all must neither be euler nor all non-euler) (2+1+2+2)
a) Figure out Euler graph from these three graphs.
b) Write down the Euler path of these graphs.
c) If not Euler, provide the reason.

Q2 Draw any three graphs (Take help from book, but DO NOT copy paste any graph from examples or exercise. Your graphs must be random and all must neither be euler nor all non-euler)
a) Figure out Euler graph from these three graphs.
b) Write down the Euler path of these graphs.
c) If not Euler, provide the reason.

Q3 X people are chosen from a volley ball team (Take a value of X by yourself, possibly that number must be close to a number of volley ball team members).

Problem B
Show that 3 · 4n + 51 is divisible by 3 and 9 for all positive integers n.