Notes:

. To obtain full marks for each question, relevant and clear steps must be included in the answers.

. Partial marks maybe awarded depending on the degree of completeness and clarity.

QUESTION I: Functions and PHP (18 marks)

1). Let  f: ℚ → ℚ , check whether f(x) = and f−1(x) = are inverses. (2 marks)

2). Determine whether the function f: ℤ  × ℤ   →  ℤ,  f(m, n) = m − n   is one-to-one, onto, or both. Prove your answer.                               (4 marks)

3). Let  S  = {x|x ∈ ℝ  and x  ≥  1} and  T  = {x|x  ∈ ℝ and 0 < x ≤ 1}.

Find a function f: S → T that is a bijection.       (4 marks)

4). A class of 32 students is organized in 33 teams. Every team consists of three students and there are no identical teams.

Show that there are two teams with exactly one common student.                           (4 marks)

5). Find the composite function g ∘ f, where:

f: ℝ → ℝ, f(x) =  {x x3(−),2,

if x ≥ 1

if x < 1

g: ℝ → ℝ,

g(x) =  { |,

if x ≥ 0

if x < 0

(4 marks)

QUESTION II: Logic (28 marks)

1). We define the following predicates:

-   F(x): xis Female

-   S(x): x is a Student

-   K(x, y): xknows y’s name

Translate the following sentences into formulas; that is,for each of the following sentences provide a formula.

a) Laura knows the name of every female student.                              (3 marks)

b) Of all pairs of students that know each other’s names, at least one in each pair is female. (3 marks)

2). If statement q has the truth value 1 (TRUE), determine all truth value assignments for

the statements p,r, and s for which the true value of the statement

(q → ((¬p ∨ r) ∧ ¬s)) ∧ (¬s →  (¬r ∧ q))

is 1 (TRUE).                     (4 marks)

3). Using  the predicate  symbols shown and appropriate quantifiers, write each English

language statement as a predicate formula. (The domain of discourse is the whole world.)

D(x): x is a day

S(x): xis sunny

R(x): xis rainy

M: Monday

T: Tuesday.

(a) All days are sunny. (2 marks)

(b) Some days are not rainy. (2 marks)

(c) It is always a sunny day only if it is a rainy day. (2 marks)

(d) Monday was sunny; therefore, everyday will be sunny. (2 marks)

(e) If someday is rainy, then everyday will be sunny. (2 marks)

4). Without using any truth tables, prove or disprove that 军 (p Λ q) V (军 p Λ q) 三 军 p.   (2 marks)

5). Translate the following first-order formulae into English and determine which of them represent true propositions when interpreted in R (the set of all real numbers).

(i)     ∀x(x > 0 → x2 > x)                                                                                            (2 marks)

(ii)    ∀x ∀y(x > y → ∃z(x > z Λ z > y)                                                                    (2 marks)

(iii)    ∀x(x ≥ 0 → ∃y(y > 0 Λ x = y2  ))                                                                   (2 marks)

We use the mathematical symbols:

=            to express the equality of the elements

> ( ≥ )    to express greater than (or equal to)

xn                   to express x raised to then power

3).      m boys and n girls are to be arranged in a row, where m, n ∈ N - {0}.

Find the number of ways this can be  done in  each of the following cases:

(i) There are no restrictions. (3 marks)

(ii) No boys are  adjacent (m ≤ n + 1). (3 marks)

(iii) Then girls form a single block. (3 marks)

(iv) A particular boy and a particular girl must be adjacent. (3 marks)

d). Suppose the council contains eight men and seven women.

(i) How many committees of six contain three men and three women? (2 marks)

(ii) How many committees of six contain at least one woman? (2 marks)

e). Suppose the council consists of three freshmen, four sophomores, three juniors, and five seniors. How many committees of eight contain two representatives from each class?   (2 marks)

QUESTION IV: Probability (16 marks)

1. Two dice are thrown. Let Ebe the event that the sum of the dice is odd, let Fbe the event that at least one of the dicelandson 1, and let G be the event that the sum is 5. Describe the events:

(i)    E “ F                                                                                                                               (2 marks)

(ii)   E  ∪F                                                                                                                            (2 marks)

(iii)  F “ G                                                                                                                              (2 marks)

2). An elementary school is offering 3 language classes: one in Spanish, one in French, and

one in German.

The classes are open to any of the 100 students in the school. There are 28 students in the Spanish class, 26 in the French class, and 16 in the German class. There are 12 students that are in both Spanish and French, 4 that are in both Spanish and German, and 6 that are in both

French and German. In addition, there are 2 students taking all 3 classes.

(i) If a student is chosen randomly, what is the probability that he or she is not in any of the language classes?               (2 marks)

(ii) If a student is chosen randomly, what is the probability that he or she is taking exactly one language class?                       (2 marks)

(iii) If 2 students are chosen randomly, what is the probability that at least 1 is taking a language class?                (2 marks)

3). A man fires at a target n = 6 times and hits it k = 2 times,

(a) List the different ways that this can happen.

(Hint: You can useS - for success and F – for failure). (2 marks)

(b) How many ways are there? (2 marks)