Definition. A set
is a collection of objects. We call the objects, elements. A set is denoted by
listing the elements between braces. For example: {e, ı, π, 1} is the
set of the integer 1, the pure imaginary number ı = √−1 and the
transcendental numbers e = 2.7182818... and π = 3.1415926.... For elements of a
set, we do not count multiplicities. We regard the set {1, 2, 2, 3, 3, 3} as
identical to the set {1, 2, 3}. Order is not significant in sets. The set {1, 2,
3} is equivalent to {3, 2, 1}.
In
enumerating the elements of a set, we use ellipses to indicate patterns. We
denote the set of positive integers as {1, 2, 3...}. We also denote sets with
the notation {x | conditions on x} for sets that are more easily
described than enumerated. This is read as “the set of elements x such
that ...”. x ∈S is the notation for “x is an element
of the set S.” To express the opposite we have x ∉ S for “x
is not an element of the set S.”
Examples. We
have notations for denoting some of the commonly encountered sets.
- ∅ = {} is
the empty set, the set containing no elements.
- Z = {...,−3,−2,−1,0,1,2,3...} is the set of integers.
(Z is for “Zahlen”, the German word for “number”.).
- Q = {p/q|p, q ∈ Z, q
≠ 0} is the set of rational numbers. (Q is for quotient). (1 Note that with this description, we enumerate
each rational number an infinite number of times. For example: 1/2 = 2/4 = 3/6 = (– 1)/(– 2) = · · · . This does not pose a
problem as we do not count multiplicities.)
- R = {x|x = a1a2···an.b1b2···}
is the set of real numbers, i.e. the set of numbers with decimal
expansions. (2 Guess what R is for.)
- C = {a + ıb|a,b ∈ R,ı2
= −1} is the set of complex numbers. ı is the square root of −1.
(If you haven’t seen complex numbers before, don’t dismay. We’ll cover them
later.)
- Z+, Q+ and R+
are the sets of positive integers, rationals and reals, respectively. For
example, Z+ = {1, 2, 3,...}. We use a − superscript to denote the
sets of negative numbers.
- Z0+, Q0+ and R0+
are the sets of non-negative integers, rationals and reals, respectively. For
example, Z0+ = {0, 1, 2,...}.
- (a...b) denotes an open interval
on the real axis. (a...b) ≡{x|x ∈ R, a
< x < b}
- We use brackets to denote the closed
interval. [a..b] ≡{x|x ∈ R, a
≤ x ≤ b}
The cardinality
or order of a set S is denoted |S|. For finite sets, the
cardinality is the number of elements in the set. The Cartesian product
of two sets is the set of ordered pairs:
X × Y
≡ {(x, y)|x ∈ X, y ∈ Y}.
The
Cartesian product of n sets is the set of ordered n-tuples:
X1 × X2
×···× Xn ≡ {(x1, x2,...,xn)|x1
∈ X1, x2 ∈ X2,...,xn
∈ Xn}.
Equality.
Two sets S and T are equal if each element of S is
an element of T and vice versa. This is denoted, S = T.
Inequality is S ≠ T, of course. S is a subset of T, S ⊆ T,
if every element of S is an element of T. S is a proper
subset of T, S ⊂ T, if S ⊆ T
and S ≠ T. For example: The empty set is a subset of every set, ∅⊆ S.
The rational numbers are a proper subset of the real numbers, Q ⊂ R.
Operations.
The union of two sets, S ∪ T,
is the set whose elements are in either of the two sets. The union of n
sets,
is the
set whose elements are in any of the sets Sj. The intersection
of two sets, S ∩ T, is the set whose elements are in both of the
two sets. In other words, the intersection of two sets in the set of elements
that the two sets have in common. The intersection of n sets,
is the
set whose elements are in all of the sets Sj. If two sets have no elements in
common, S ∩ T = ∅, then the sets are disjoint. If T ⊆ S, then
the difference between S and T, S \T, is the set of elements in S which are not
in T.
S \ T
≡ {x|x ∈ S, x ∉ T}
The
difference of sets is also denoted S − T.
Properties.
The
following properties are easily verified from the above definitions.
- S ∪ ∅ = S, S ∩ ∅ = ∅, S
\∅ = S,
S \S = ∅.
- Commutative. S ∪ T
= T ∪ S, S ∩ T = T ∩ S.
- Associative. (S ∪ T)
∪ U
= S ∪ (T ∪ U)
= S ∪ T ∪ U,
(S ∩ T) ∩ U = S ∩ (T ∩ U) = S
∩ T ∩ U.
- Distributive. S ∪ (T
∩ U) = (S ∪ T) ∩ (S ∪ U),
S ∩ (T ∪ U) = (S ∩ T) ∪ (S
∩ U).
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