3. Set theory is the branch of mathematics that studies
sets, which are collections of objects. Although any
type of object can be collected into a set, set theory is
applied most often to objects that are relevant to
mathematics.
The modern study of set theory was initiated by
Cantor and Dedekind in the 1870s. After the discovery
of paradoxes in informal set theory, numerous
axiom systems were proposed in the early twentieth
century, of which the Zermelo–Fraenkel axioms, with
the axiom of choice, are the best-known.
4. Set theory begins with a fundamental binary relation between an
object o and a set A. If o is a member (or element) of A, we
write . Since sets are objects, the membership relation can relate
sets as well.
A derived binary relation between two sets is the subset relation,
also called set inclusion. If all the members of set A are also
members of set B, then A is a subset of B, denoted . For
example, {1,2} is a subset of {1,2,3}, but {1,4} is not. From this
definition, it is clear that a set is a subset of itself; in cases where
one wishes to avoid this, the term proper subset is defined to
exclude this possibility.
5. Just as arithmetic features
binary operations on numbers, set theory
features binary operations on sets. The:
1) Union of the sets A and B, denoted , is the
set whose members are members of at least one of A
or B. The union of {1, 2, 3} and {2, 3, 4} is the set {1,
2, 3, 4}.
6.
7. 3) Complement of set A relative to set U, denoted , is the set
of all members of U that are not members of A. This terminology
is most commonly employed when U is a universal set, as in the
study of Venn diagrams. This operation is also called the set
difference of U and A, denoted The complement of {1,2,3}
relative to {2,3,4} is {4}, while, conversely, the complement of {2,3,4}
relative to {1,2,3} is {1}.
8. •Symmetric difference of sets A and B is
the set whose members are members of
exactly one of A and B. For instance, for the
sets {1,2,3} and {2,3,4}, the symmetric
difference set is {1,4}.
9. The power set of a
set Ais the set
whose members are
all possible subsets
of A For example,
.
the power set of { 1,
2} is { { } , { 1} , { 2} ,
{ 1,2} } .
10.
11. In this we define a set by actually
listing its elements, for example , the
elements in the set A of letters of the
English alphabet can be listed as
A={a,b,c,……….,z}
NOTE: We do not list an element more
than once in a given set
12. In this form,set is defined by stating properties which the
statements of the set must satisfy.We use braces { } to write
set in this form.
The brace on the left is followed by a lower case italic letter
that represents any element of the given set.
This letter is followed by a vertical bar and the brace on the
left and the brace on the right.
Symbollically, it is of the form {x|- }.
Here we write the condition for which x satisfies,or more
briefly, { x |p(x)},where p(x) is a preposition stating the
condition for x.
The vertical is a symbol for ‘such that’ and the symbolic form
A={ x | x is even } reads
“A is the set of numbers x such that x is even.”
Sometimes a colon: or semicolon ; is also used in place of the
13. A set is finite if it consists of a
definite number of different elements
,i.e.,if in counting the different
members of the set,the counting
process can come to an end,otherwise
a set is infinite.
For example,if W be the set of people
livilng in a town,then W is finite.
If P be the set of all points on a line
between the distinct points A and B
14. A set that contains no members is called
the empty set or null set .
For example, the set of the months of a
year that have fewer than 15 days has
no member
.Therefore ,it is the empty set.The empty
set is written as { }
15. Equal sets are sets which have the
same members.For example, if
P ={1,2,3},Q={2,1,3},R={3,2,1}
then P=Q=R.
16.
17.
18. (1) EvEry sEt is a subsEt of itsElf.
(2) thE Empty sEt is a subsEt of EvEry
sEt.
(3)
