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Unformatted text preview: SIT192 Assignment 1
Due: Friday March 25th, 2011
1. The propositions p, q , r, and s are deﬁned by:
p: they eat
q : they drink
r: they read
s: they study
Express the following in symbolic form:
(i) They don’t drink, but they read and study.
(ii) If they study, they eat and drink.
(iii) They don’t eat if they read and don’t drink.
(iv) If they read and don’t study, they neither eat nor drink.
2. Using the notation in Question 1, translate the following expressions into
English.
(i) q ∨ (s ∧ r)
(ii) ¬r → ¬p
(iii) (s ∧ r) ↔ ¬(p ∨ q ) (iv) ¬p → (r ∧ ¬q )
3. Construct the truth tables for the following expressions.
(i) (p ∨ ¬q ) ∧ q (ii) ¬p → ¬(p ∧ q )
(iii) (p ∧ q ) ↔ ¬r
4. Use the algebra of propositions to prove.
(i) p ∨ ¬(p ∨ q ) ≡ p ∨ ¬q (ii) p ∧ ¬(¬q ∧ r) ≡ (p ∧ q ) ∨ (p ∧ ¬r)
(iii) (p ∧ q ) → p ≡ T
(iv) q → (p ∧ q ) ≡ q → p
5. Determine the truth value of each of the following statements, where x and
y are real numbers.
(i) ∃x : 3x2 − 1 ≥ −6
(ii) ∀x : 5 − 2x2 ≥ 7
(iii) ∀x∃y : y = x + 9
(iv) ∃y ∀x : y = x + 9
2
2
2
(v) ∃y ∀x : (x + 3y ) = x + 3x + 9y .
1 6. Given that n is an integer, prove that n is even if and only if 9n2 + 4n − 11
is odd.
7. The universal set U is given by U = {x ∈ N : 22 ≤ x ≤ 42}, where N is the
set of natural numbers. Given that
A = {x ∈ U : x > 37},
B = {x ∈ U : x is a multiple of 4}, and
C = {x ∈ U : x is a multiple of 7}.
List the elements of the following set. (i)
A
(ii)
B
(iii) C
(iv) A ∩ B (v)
B ∪ C (vi) A ∩ B
(vii) C − B (viii) B ⊕ C (ix) (A − B ) ∩ C
8. For the sets A, B , and C deﬁned in Question 7, decide whether each of the
following statements is true or false.
(i) 41 ∈ B
(ii) B ∩ C = {28}
(iii) {42}A ⊆ B ∩ C (iv) 23 ∈ (A ∩ B )
(v) {35} ∈ C − B
(vi) B ∩ C ⊆ A
9. Use the laws of set algebra to prove
(i) A ∪ (A ∪ B ) = A ∪ B
(ii) (A ∪ B ) − A = A − B. 2 SIT192 Assignment 2
Due: Friday April 15th, 2011
1. Given that Z is the set of integers, R is the set of real numbers, determine
whether each of the following functions is a bijection:
(i) f : R → R, where f (x) = 5x + 14. (ii) f : Z → R, where f (x) = 7 x− 2
.
5 (iii) f : R → R, where f (x) = (5x + 8)2 .
2. Use the bubble sort to arrange the letters Z , Q, B , S , A, L into alphabetical
order, showing the lists obtained at each step.
3. Prove, by ﬁnding witnesses that:
(i) f (x) = 15x4 − 8x2 + 19x − 4 is O(x4 ). (ii) f (x) = 10x3 + log2 x − 5x + 3 is O(x3 ).
4. Find the prime factorisation of each of the following integers.
(i) 58 (ii) 97 (iii) 165 (iv) 391 (v) 902
5. Find the greatest common divisors and the least common multiples for each
fo the following pairs of integers:
(i) 34 and 49
(ii)
8
3
4
(iii) 3 × 7 × 13 × 23
and 87 and 203
25 × 36 × 7 × 192 6. Find:
(i) 523 mod 26; (ii) −191 mod 11; (iii) −499 mod 53;
(iv) 9602 mod 77, given that 915 ≡ 1 mod 77.
7. (a) Convert the following integers from the given base to Base 10 notation:
(i) (583)9 ; (ii) (1403)5
1 (b) Convert 423 from Base 10 notation to:
(i) Base 6 notation; (ii) Base 8 notation.
8. Use the Euclidean Algorithm to ﬁnd the greatest common divisors of each
of the following pairs of integers.
(i) 171 and 456 (ii) 159 and 690. 9. Given that A = 4 −1
21 0 −1 and B = 3 −2 52 (i) AAT (ii) AT A
(iii) BA (iv) BB T
(v) A2 (vi) A3 2 SIT192 Assignment 3
Due: Friday May 13th, 2011
1. Use mathematical induction to prove the following:
(i) 13n − 1 is divisible by 4, for all integers n ≥ 0. (ii) 2n3 − 3n2 + 5n − 16 ≥ 0, for all integers n ≥ 3.
2. How many positive integers between 200 and 799 inclusive that:
(i) have no two digits the same;
(ii) are divisible by 7;
(iii) are divisible by 23;
(iv) are divisible by 7 and by 23;
(v) are divisible by 7 or by 23; and
(vi) are divisible by 7 but not by 23.
3. Given that A = {1, 2, 3} and B = {1, 2, 3, 4, 5, 6, 7}, how many functions
f : A B are there
(i) in total?
(ii) which are onetoone?
(iii) with f (2) = 2, and f (3) = 4?
(iv) with f (1) = 2, and f (3) = 7? 4. Find the minimum number of people required to ensure that:
(i) 23 were born on the same day of the week; and
(ii) 19 have family names which start with the same letter of the alphabet.
5. A board of 10 people is to be selected from a panel of 11 women and 10 men.
In how many ways can the board be chosen if it must contain
(i) exactly 2 women?
1 (ii) at least 2 women?
(ii) a male president of the board, in addition to 2 women and 7 men?
6. Find the coeﬃcient of x27 y 3 in the expansion of (x − 3y )30 .
7. Find the number of nonnegative integer solutions to:
x1 + x2 + x3 + x4 + x5 + x6 = 3
8. Using all the letters, ﬁnd the number of diﬀerent strings that can be made
from the following words:
(i) ARARAT (ii) ORROROO (iii) TALLANGATTA
9. For each of the following relations R, on the set A = {1, 2, 3, 4}, determine
(and explain) whether R is
(a) reﬂexive (b) symmetric (c) transitive
(i) R = {(1, 1), (1, 2), (3, 4)}; (ii) R = {(1, 1), (2, 2), (2, 3), (3, 2)}; and (iii) R = {(1, 1), (2, 1), (2, 2), (3, 3), (4, 4)}. 2 SIT192 Assignment 4
Due: Friday May 27th, 2011
1. For each of the following relations R on the set A = {1, 2, 3, 4}:
(a) ﬁnd the matrix representing R;
(b) draw the direction graph corresponding to R.
(i) R = {(1, 1), (1, 4), (2, 3), (2, 4), (3, 3), (4, 1), (4, 2), (4, 3)}.
(ii) R = {(1, 1), (1, 3), (1, 4), (2, 2), (3, 1), (3, 3), (3, 4)}.
(iii) R = {(a, b) : a + b is a multiple of 3}.
2. Given the following relations R on the set of nonnegative integers, test
whether R satisﬁes each of the properties of an equivalence relation.
(i) R = {(x, y ) : x ≡ y mod 8}. (ii) R = {(x, y ) : x + y is an odd number }.
3. Find the equivalence classes for each of the following equivalence relations R
on the given sets A.
(a) R = {(x, y ) : x ≡ y mod 8}, where A is the set of nonnegative
integers;
(b) R = {(1, 1), (2, 2), (2, 4), (3, 3)(4, 2), (4, 4)}, where A = {1, 2, 3, 4}.
4. (a) Draw the graph representing the following mutual acquaintanceships
between the seven people A, B, C, D, E, F, and G:
A knows B, C, D, E, and F;
B knows C and E;
C knows D, E, and F;
D knows E and G;
E knows F and G;
(b) Verify that the Handshaking Theorem holds for the graph in Part (i). 1 5. (a) Draw the following graphs.
(ii) K7 (ii) K1,6 (iii) K2,5
(iv) C9 (v) W9
(b) Verify that the Handshaking Theorem holds for each of the graphs in
Part (a).
6. (a) Draw the graphs on 3 vertices a, b, and c whose adjacency matrices are: 011
122
(i) A = 1 1 0 ; (ii) B = 1 0 2 111
231
(b) Find the number of paths of length 2 and length 3 from vertex a to
vertex c for each of the graphs in Part (a). 2 ...
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