MATB24H3 Lecture Notes - Lecture 1: Empty Set, Rational Number, If And Only If

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Published on 4 Oct 2020
Aset is a container with no distinguishing feature other than its contents. The objects
contained in a set are called the elements of the set. We write aSto signify that the object a
is an element of the set S. The number of elements in a set Sis called the cardinality of the set,
and it is denoted by |S|.
Since a set has no distinguishing feature other than its contents, there is a unique set containing
no elements which is called the empty set and is denoted . Some other very common sets are
the set Rof all real numbers, the set Qof all rational numbers, the set Zof all integers, and the
set Cof all complex numbers.
There are two important ways to specify a set.
Enumeration. One can list the contents of the set, in which case the set is denoted by
enclosing the list in curly braces. For example, Z={. . . , 2,1,0,1,2, . . .}.
Comprehension. One can describe the contents of the set by a property of its elements.
If P(a)is a property of the object a, then the set of all objects asuch that P(a)is true
is denoted by {a|P(a)}, or equivalently {a:P(a)}. For example,
bfor some a, b Zwith b6= 0}.
Let Xand Sbe sets. We say that Sis a subset of Xif aS=aXholds for all objects
a. We write SXto signify that Sis a subset of X. This means that Sis a set each of
whose elements also belongs to X. The subset of Xconsisting of all elements aof Xsuch that
property P(a)holds true is denoted {aX|P(a)}.
EXERCISE 1.1. (1) Give common English descriptions of the following sets:
(i) {nN|for all mN, if nis a multiple of mthen m= 1 or m=n}.
(ii) {zC|z=x2for some xR}.
(2) Use set comprehension notation to give a description of each of the following sets:
(i) The set of all integers that are powers of prime numbers.
(ii) The plane consisting of all points in R3whose coordinate entries sum to 1.
(3) Let S={a, b, c}. One subset of Sis the set {a, b}, so we may write {a, b} ⊆ {a, b, c}.
List all eight of the subsets of S.
Starting from given sets, we can use set operations to form new sets.
Given sets Xand Y, the intersection of Xand Yis defined as
XY={a|aXand aY}.
Given sets Xand Y, the union of Xand Yis defined as
XY={a|aXor aY}.
Given sets Xand Y, the difference of Xand Y, denoted X\Yor XY, is the set
{xX|x /Y}.
Given a set Yinside some larger set X, the complement of Ywith respect to X,
denoted YC, is X\Y. (The larger set X, sometimes referred to as the universe, is
often suppressed in the notation).
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