CS计算机代考程序代写 flex /Users/billy/gits/moc-2021/tutes/week07.dvi

/Users/billy/gits/moc-2021/tutes/week07.dvi

School of Computing and Information Systems

COMP30026 Models of Computation Tutorial Week 7

6–10 September 2021

Content: sets, relations, properties of relations, functions

The exercises

T7.1 Let A = {1, 2}, B = {2, 3, 4} be sets, R1 = {(1, 4), (2, 3)} a relation from A to B and
R2 = {(3, 1), (2, 2)} a relation from B to A. Evaluate the following sets in list form.

(i) (A ∪B) \ (A ∩B)

(ii) A× P(B)

(iii) R1 ◦R2

(iv) (R1 ◦R2) ∩ P(A)

T7.2 Let A, B be any sets. Show the following, using the logical definitions of the relevant concepts.

(i) A 6⊆ B ⇔A \B 6= ∅. (ii) A ∩B = A \ (A \B).

T7.3 We can represent a relation R ⊆ A × A as a directed graph, where the vertices are elements
of A and there is a directed edge from x to y, iff (x, y) ∈ R. Below is a graph representing a
relation R on the set {1, 2, . . . , 8}.

1

2

3

4

5

67

8

Draw the graph representing the transitive closure of R. Is it reflexive? Is the symmetric-
transitive closure of R reflexive?

T7.4 Show that a relation R on A is transitive iff R ◦R ⊆ R. Then give an example of a transitive
relation R for which R ◦R = R fails to hold.

T7.5 Suppose we know about functions f : A → B and g : B → A that f(g(y)) = y for all y ∈ B.
What, if anything, can be deduced about f and/or g being injective and/or surjective?

T7.6 Suppose h : X → X satisfies h ◦ h ◦ h = 1X . Show that h is a bijection. Also give a simple
example of a set X and a function h : X → X such that h ◦ h ◦ h = 1X , but h is not the
identity function (hint: think paper-scissors-rock).

After you finish: work on questions from Problem Set 7