What are openand closed sets?
Formally, an open setis one in which every element in the set has a neighborin the
set or does not include its boundary. A closed setis one that does include a boundary
of a set, or one where some elements have neighbors not in the set.What is a nullor empty set?
A null or empty set contains no elements; an empty set is considered to be a subset of
every other set. The opposite of an empty set is, logically, a nonemptyset, or one that
is not empty. The notations for empty set are {} and , but not (), as it is sometimes
written in texts. Interestingly enough, an empty set is considered to be both open and
closed for any set X.What is the set theory approachto arithmetic?
The set theory approach to arithmetic is defined in terms of the nonnegative whole
numbers 0, 1, 2, 3.... These numbers are identified with specific sets based on the
placement and number of brackets. For example, mathematicians identify 0 with the
empty set { }; 1 is identified with {{ }}; 2 is identified with {{}, {{}}}; 3 is identified with
{{ }, {{ }}, {{ }, {{ }}}}, and so on (each bracket is an interpretation of the empty set and
1, or {{}}).What are the basic symbolsused to operate on sets?
When set theory founder Georg Cantor developed the symbols for sets, he used a sin-
gle horizontal overbar to denote a set with no structure besides order; thus, it repre-
sented the order type of the set. A double bar meant that there was no order from the
set, which is also called the cardinal number of the set (see below).
The more common symbols accepted in today’s set theory include the following:Symbols Used to Operate Sets
Symbol Meaning
“and” or intersection (the intersection of two sets)
“or” or union (the union of two sets)
set membership (such as a Sin which ais an element of the set S)
and subset (such as ABor every element of Ais also an element of B)
not a set membership (such as a S, in which ais not an element of S;
this symbol is larger and thicker than set membership
empty or null set (a set with no elements)
{} braces or brackets (to enclose the listed elements of a set; empty
124 brackets are called an “empty set,” or a set with no elements)