G is strongly connected graph

A. Let A = x ∈ N

                  B = x = 5a, for some a from N and 10 ≤ x ≤ 20

1. Describe each of A and B using roster method
2. Find A ∪ B,  A ∩ B , A – B, B – A, and A ⊕ B  then use set builder method to describe  A ∩ B
3. Find |A ∪ B| and P (A ∩ B)

B. Why is f not a function from R to R if

1. f(x) = 1 / x
2. f(x) = √x
3. f(x) =  ±√(x^2 + 1)

C. Determine with proper justification whether each of these functions is a one to one, onto ,or a bijection from R to R

1. f(x) = 3x – 2
2. f(x) = x^2 – 1

1.

Let A be the set A = 1, 2, 3, 4 and R a relation on A defined as the following:

R = (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 4), (3, 1), (3, 3), (4, 1), (4, 2), (4, 4)

Answer all the following questions:

1. Represent R with a matrix (considering the elements of the set A listed in the same order as above) 
2. Determine whether R is reflexive, symmetric, antisymmetric or transitive
3. Determine which of the following graphs represents R

2. Consider this directed graph G and answer the following questions:

A.   Determine whether these sequences of vertices are paths in this directed graph G.

a) a, b, d, e

b) b, d, c, b, f

c) a, a, b, d, e, e

d) b, c, e, d, a, b

e) e, a, c, c, b, d

B. Determine whether G is strongly connected graph, and find deg-(a), deg-(e), deg+(c), deg+(d)

C.  List two circuits of length four in G.

3. Construct the adjacency matrix for G considering the alphabetic order for its vertices.

Formulate the Boolean expression that represents the output  of each of  the following circuits

Use  the numbers (1, 2 , 3, ..) to express the output of  the corresponding gates / components as illustrated in the below figures