2.3. Congruences to a modulus. It is sometimes convenient to identify two numbers
which differ by an integer multiple of some given number. For example, two events happen
at the same time of day if their times differ by an integer multiple of 24 hours.
Le
Mathematics 2F, Solutions to exercises, 20162017
IV. Isometries of the plane.
1. (a) Let Q be the projection of P (2, 3) on L. Then Q has coordinates
(2, 3) + t(1, 2) = (2 + t, 3 + 2t).
We have
2 + t + 2(3 + 2t) = 2,
hence t =
56 .
The reflection R of P
Mathematics 2F, Solutions to exercises, 20162017
I. Sets, functions, cardinality and equivalence relations
1. Suppose firstly that x
/ A B. Then it is not true that x belongs to both A and B. Then x
does not belong to at least one of these sets A, B. Hen
2. Number theory
2.1.
Greatest common divisors and the Euclidean algorithm.
Number theory is the study of the integers, especially the positive integers. Positive
integers are called natural numbers. The set consisting of positive integers is
denoted N, a
Definition 3.21. Let g be an element in a group G. The the subgroup of G
generated by g, denoted hgi, is the subgroup given by
hgi= cfw_g n : n Z .
The order of g, denoted |g|, is the order of the subgroup hgi.
Example 3.22. The subgroup of Z generated by
1. Sets, functions, cardinality and equivalence relations
1.1. Sets. A set S is a collection of objects called the elements or members
of S.
One writes x S to mean that x is a member of S and x / S to mean that
x is not a member of S.
One writes
cfw_x , x
1.3. Finite sets. We consider the sizes of sets.
For n = 0, 1, 2, . . . let
n = cfw_1, 2, . . . , n,
so that, in particular, 0 = ; thus n is a standard set with n elements.
Definition 1.20. Let X be a set. If there is a bijection n X for some
nonnegative
1.2. Functions.
Definition 1.7. A function or mapping f from a set X to a set Y consists of
a rule assigning a member f (x) of Y to each member x of X .
Suppose that f is a function from X to Y . We write f : X Y , and we write
x 7y if y is the element f
1.4. Countable sets. We will now consider the smallest infinite sets. We
begin by considering the subsets of the set of natural numbers
N = cfw_1, 2, 3, . . . .
Proposition 1.27. The set N is infinite. If E is a subset of N, then E is finite
or there is a
1.5. Uncountable sets. Although many apparently large infinite sets are in fact
countable, there are also uncountable sets, which can be constructed from
power sets. Recall that the power set P(X ) of a set X is the set whose members
are the subsets of X
Both of the permutations are odd.
Definition 3.9. Let n be a nonnegative integer. Then the alternating group
on n objects, denoted An , is the subset of Sn consisting of the even permutations.
For example,
A0 = cfw_Id= S0 , A1 = cfw_Id
= S1 , A2 = cfw_Id,
3. Permutations and groups
3.1. Permutations. Let X be a set. A permutation of X is a bijective func- tion from
X to X . The set of permutations of X is denoted Perm(X ).
Proposition 3.1. Let X be a set.
(i) If f, g Perm(X ) then there is a composite f g
Mathematics 2F, Solutions to exercises, 20162017
II. Number theory
1. (a) If a | b and b | c, then b = ua and c = vb with u, v Z, hence
c = vb = v(ua) = (vu)a
with vu Z, hence a | c.
(b) Suppose that a | b and b | a. Then there are integers r, s such that