1) Let f (x) = x2 + 1 and g(x) = x + 2 are relations on the set of real numbers.

(a) Decide whether f(x) and g(x) are function, if so, state whether they are injection,

surjection or bijection. (4 marks)

(b) Does either f (x) or g(x) have an inverse? If so, find the inverse. (3 marks)

(c) Find f ◦ g and g ◦ f . (6 marks)

2) For each of the following relations on the set {1, 2, 3, 4}, decide whether it is reflexive,

whether it is symmetric, whether it is antisymmetric, and whether it is transitive:

(6 marks)

a) {(2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)}

b) {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}

c) {(2, 4), (4, 2)}

3) Let A = {7,9,10,12}, B = {1,2,3,4} and consider the relations:

R = {(a,b)ÎA´B | a-2b ³ 3}

G = {(a,b) | aÎA , bÎB and (a+b) = 12}

H = {(a,b) | aÎB , bÎB and (a+b) > 5}

(a) List all the elements of the relation R and give its cardinality |R|. (4 marks)

(b) Find the range and domain of the relation R. (2 marks)

(c) Find the inverse relation R-1. (2 marks)

(d) Find the sets of ordered pair in G and H. (6 marks)

(e) Draw the diagraphs of G and H. (6 marks)

(f) Give the Boolean-matrix representations of G and H. (2 marks)

4) Determine whether each of these functions is a bijection from ℝ to ℝ, where ℝ is the set

of real numbers, justify your answers: (6 marks)

(a) f(x)=2x+1

(b) f(x)=x2 +1

(c) f(x)=x3