Tutorial Sheet 4
Aug 17, 19, 20
1. Determine whether each of these sets is countable or uncountable. For those that are countably, exhibit a one-to-one
mapping from the set to the set of positive integers.
(a)
(b)
(c)
(d)
(e)
(f)
all positive rational num

Tutorial Sheet 8
Sept 28, 30, Oct 1
1. How many n-digit decimal sequences (using the digits 0 = 9) are there in which the digits 1, 2 and 3 all appear?
2. How many arrangements of 52 letters, 2 As, 2 Bs, 2 Cs, etc. are there with no pair of consecutive le

Tutorial Sheet 7
Sept 21, 23, 24
1. Let n be a positive integer and let n 1 = 2s t , where s is a nonnegative integer and t is an odd positive integer. We say
j
that n passes Miller's test for the base b if either bt 1( mod n) or b2 t 1( mod n) for some j

Tutorial Sheet 9
Oct 5,6,7
1. Find a recurrence relation for the number of ternary strings of length n
that contain either two consecutive 0s or two consecutive 1s.
2. Find a recurrence relation for the number of bit strings of length n that
contain the s

Tutorial Sheet 6
Sept 6,8,9
1. Show that if a and b are both positive integers, then (2a 1) mod (2b 1) = 2a
mod b
1.
2. Use the above to show that if a and b are positive integers, then gcd(2a 1, 2b 1) = 2gcd(a,b) 1. [Hint: Show that
the remainders obtai

Tutorial Sheet 5
August 24, 26 and 27
1. A drawer contains a dozen brown socks and a dozen black socks, all unmatched. A man takes socks out at random in
the dark.
(a) How many socks must he take out to be sure that he has at least two socks of the same c

Tutorial Sheet 6
Sept 6,8,9
1. Show that if a and b are both positive integers, then (2a 1) mod (2b 1) = 2a
mod b
1.
2. Use the above to show that if a and b are positive integers, then gcd(2a 1, 2b 1) = 2gcd(a,b) 1. [Hint: Show that
the remainders obtai

Tutorial Sheet 3
Aug 10, 12, 13
1. Let a1 , a2 , ., an be positive real numbers. The arithmetic mean of these numbers is dened by A = (a1 +a2 +an )/n,
and the geometric mean of these numbers is dened by G = (a1 a2 . . . an )1/n . Use mathematical inductio

Tutorial Sheet 2
Aug 3, 5, 6
1. Establish these logical equivalences, where x does not occur as a free variable in A. Assume that the domain is
nonempty.
a. x (A P(x) ()
b. x(A P(x) ()
2. Find a common domain for the variables x, y, and z for which the st

Tutorial Sheet 1
27, 29, 30 July
1. Determine whether ( ( ) ) is a tautology.
2. Show that ( ) and are logically equivalent.
3. The nth statement in a list of 100 statements is Exactly n of the statements in this list are false.
a. What conclusions can yo