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Exercise 1.

Let A = fx; y; zg. (i) Give an example of three elements from the Cartesian product P(A) A.

(ii) How many elements are in the Cartesian product P(A) A.

Justify your answer.

Exercise 2.

In each of the following statements replace `?’ by the element such that the statement is true. If there is no solution give a reason. If there are more solutions, give all of them.

(i) f?; ?; ?g 2 f1; 2; 3; f5; 6; 7gg

(ii) ff?; ?gg f1; 2; f3; 4gg

(iii) f?; ?g f1; 2; 3; f4; 5gg

Exercise 3. Give an example of a formula which describes the following innite set:

f11; 15; 19; 23; 27; : : : g:

Exercise 4

. (i) How many subsets of the set fa; b; c; d; eg doesn’t contain the element a?

(ii) How many subsets of the set fa; b; c; d; eg contain the elements a or b (including the subsets which contain both elements a, b)? Justify your answer.

Exercise 5.

How many elements are in the set f(a; b) j (a; b) 2 NN and 1 a b ng

for n 1, n 2 N. Justify your answer.

Exercise 6. Let A, B, and C be sets. Is it true that always

A [ (B \ C) = (A [ B) \ C ?

Either prove the statement or nd a counterexample.

Exercise 7.

There is a group of 191 students of which 10 are taking operating systems, databases, and discrete mathematics; 36 are taking operating systems and databases; 20 are taking operating systems and discrete mathematics; 18 are taking discrete mathematics-

ics and databases; 65 are taking operating systems; 76 are taking databases; and 63 are taking discrete mathematics.

(i) How many are taking discrete mathematics or operating systems (or both) but not databases?

(ii) How many are taking none of the three subjects?

Exercise 8.[hard in general case] Let n > 3, n 2 N and A = f1; 2; 3; : : : ; ng.

(i) How many subsets of A contain the elements 1 and 2? 3

(ii) How many subsets B of A have the property that B \ f1; 2g = ;?

(iii) How many subsets B of A have the property that B [ f1; 2g = A

Clearly explain your answers. If you do not know how to solve it for variable n, solve the exercise for n = 5.

Exercise 9. Let A = f1; 2; 3; 4; 5; 6g. In each case (i)-(iii) give an example of a relation on the set A with the given properties. The relations can be described by listing the ordered pairs or by a formula. Clearly explain your solution. If such a relation doesn’t

exist give a reason.

(i) Give an example of a relation on the set A with at most ve elements which is re exive, but not transitive.

(ii) Give an example of a relation on the set A with at most 3 elements which is not re exive, but symmetric and transitive.

(iii) Give an example of an equivalence on the set A. Can you describe such an equiva- lence by a formula?

Exercise 10.

[hard] Let A = f1; 2; 3; 4; 5; 6g and S = P(A) be the power set of A. For a; b 2 S, dene a relation R: (a; b) 2 R if a and b have the same number of elements. Is R an equivalence relation on S? If R is an equivalence, how many equivalence classes are there? Justify your answers.

Exercise 11.

In the question below you can describe the relations/functions either by drawing a diagram, by a formula, or by listing the ordered pairs. Explain your solution.

(i) Give an example of two sets A and B and a relation R from A to B which is not a function.

(ii) [hard] Can you nd a set A, jAj = 4 and dene a bijective function between A and

P(A)? If such a set doesn’t exist give a reason.

Exercise 12.

Given the function h : N ! N; h(x) = x2 + 10. Determine whether the

function h is . . .

{ partial/total,

{ injective,

{ surjective,

{ bijective.

In each case clearly justify your answer.

Dene an inverse function of the function h. If such a function doesn’t exist, give a reason.

Exercise 13. Let f : N ! N be dened as f(x) = 3x + 2 for every x 2 N. Calculate the following and write down either value or a function. Show your working.

4

(i) (f f)(x) = f(f(x)) =

(ii) Compose the function f n-times recurrently (|f f f{z f f})

n(x) =

Exercise 14.

(i) Give an example of a function f : N ! N which is total and injective, but not surjective. If such a function doesn’t exist, give a reason.

(ii) Give an example of a function f : N ! N which is partial, injective, and surjective. If such a function doesn’t exist, give a reason.

Exercise 15.

Determine whether the following statement is a tautology, a contingency or a contradiction. Give a reason for your answer:

(p ! q) ^ (:p ! p) ! q

Exercise 16.

Let P be the proposition \Roses are red” and Q be the proposition \Violets are blue”. Express each of the following propositions as logical expressions:

(i) If roses are not red, then violets are not blue.

(ii) Roses are red or violets are not blue.

(iii) Either roses are red or violets are blue (but not both).

Exercise 17. Verify the following equivalence by writing an equivalence proof. That is,start on one side and use known equivalences to get to the other side.

(p ! q) ^ (p _ q) q

Exercise 18.

Write each of the following statements in symbolic form and determine whether they are logically equivalent. Include a truth table and a few words of explana- tion.

\If you paid full price, you didn’t buy it at Crown Books. You didn’t buy it at Crown Books or you paid full price.”

Exercise 19. Let

s(x) denote the statement \x is a student”,

h(x) denote the statement \x is happy”.

5

Formalise each of the following statements. The domain for all variables is the set of all

people.

(i) \Some student is happy.”

(ii) \Not all students are happy.”

(iii) \Every student is happy.”

(iv) \There is a sad student.”

(v) \All students are sad.”

Exercise 20.

The following picture shows an arrangement of objects of various shapes (cycles, squares, triangles) drawn with various lines (solid, dashed, dotted), which are located on a grid. The conguration can be described using logical operators such as

Triangle(x), meaning “x is a triangle”, Dashed(y), meaning \y is drawn in dashed style”, and Above(x, y), meaning \x is above y (but possibly in a dierent column)”, . . . . Indi- vidual objects can be given names such as a, b, or c.

a b

c d

e f

g h i

j k

Determine the truth or falsity of each of the following statements. The domain for all variables in (i){(iv) is the set of objects shown above. Clearly justify your answer.

(i) 8u, Circle(u) ! Dashed(u),

(ii) 9z such that Triangle(z) ^ Above(f; z).

(iii) 8 circles x 9 a square y such that x and y are draw in the same style (solid, dotted, dashed).

Exercise 21.

Let D = E = f2;1; 0; 1; 2g. Determine the truth value of each statement and justify your answer.

(i) 8x 2 D 9y 2 E such that x + y = 2.6

(ii) 9x 2 D such that 8y 2 E x y.

Exercise 22.

Write a negation (formal or informal) for each of the following statements.Be careful to avoid negations that are ambiguous.

(i) All dogs are friendly.

(ii) Some estimates are accurate.

(iii) 8x 2 R, if x2 1 then x > 0.

Exercise 23.

Give a direct proof of the fact that a2 5a + 6 is even for any integer a.

Exercise 24.

Write down the contrapositive and the negation of the following implication.

\If x2 + x 2 < 0, then x > 2 and x < 1.”

Exercise 25.

Disprove the following statement by giving a counterexample:

8a 2 Z 8b 2 Z; if a < b then a2 < b2.

Exercise 26.

[hard] If a statement contains two dierent quantiers 8 and 9, reversing their order can change the truth value of the statement. Give an example of a predicate with two free variables such that changing the order of the quantiers changes the truth value of the statement.

Exercise 27.

[hard] Rewrite the denition of a surjective function f : A ! B using 8

and 9. Write down the negation of that denition, it means use 8 and 9 to express that`function is not surjective’.

Exercise 28.

[hard] Prove that for all integers n, n 1

1 + 3 + 5 + + (2n 1) = n2.

Exercise 29.

Draw all non-isomorphic graphs with four vertices and at most four edges.

Exercise 30.

(i) Give an example of a connected graph with at least ve vertices that has as an

Eulerian circuit, but doesn’t have a Hamiltonian cycle.

(ii) Can you nd/describe a graph with n vertices that has an Eulerian circuit, but

doesn’t have a Hamiltonian cycle?

Show that graphs from (i) and (ii) have the specied properties.

Exercise 31.

Give an example of two connected graphs with the same degree sequences that are not isomorphic. Give a reason why they are not isomorphic. 7

Exercise 32. Figure out all values of k 2 N for which there exists a connected graph with the sequence (5; 5; 5; 5; 5; 5; 5; 5; 5; k). In each case either draw a graph or explain why such graph doesn’t exist.

Exercise 33.

Decide whether the answer to each individual question is `True’ or `False’ and give a reason.

(i) The complete bipartite graph Km;n is an Eulerian graph for any m; n 2.

(ii) If G is a simple graph with at least two vertices, then it is always possible to nd

two vertices of G with the same degree.

(iii) If the degree of each vertex in a connected graph G is at least 2, then G is Hamil- tonian.

(iv) If G is Hamiltonian, then the degree of each vertex is at least 2.

Exercise 34. What is the largest possible number of vertices in a connected graph with 35 edges, all vertices having degree at least 3? Can you verify your result and nd a graph with such properties?

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