Lecture 1s: Elements of Set Theory
Lecture 1s: Elements of Set Theory
Lecture 1s: Elements of Set Theory
Contents
1 Sets and Operations on Sets
Sets
Definition 1. A set is a collection of objects of some kind with a common
property that, given an object and a set, it is possible to decide if the object
belongs to the set.
We denote elements of a set by lower case letters (e.g. x, y, z, . . . ) and sets
by upper case (e.g. X, Y, Z, . . . ).
x X means x belongs to set X
Example 2. Let A = {apple, orange, lemon}. Then orange A and cucumber
/
A.
Example 3. Let X = {x1 , . . . , x10 }. Then x2 X and x100
/ X.
Y = {1, 3, 5, 7, 9}
Set comprehension
X = {x : P (x)}
where : means such that and P (x) is a rule that describes the common
property.
Example 4. Set Y can be written
Y = {y N : x < 10 and x is odd}
Example 5.
P
= {p : p is a prime number}
= {y : y = x2 , x R}
Subsets
If every element of set A is also an element of set B, then we write A B
or A B.
subset (proper)
{a, b, c} {a, b, c, d}
subset or equal
{a, b, c} {a, b, c}
equivalence of sets
AB
Remark 1.
is
A B and A B
Special Sets
L. Kronecker:
God gave us the integers; the rest is the work of Man
= {} an empty set
N = {1, 2, . . . } set of natural numbers
Z = {0, 1, 2, . . . } set of integer numbers
Q = {0, 1, 21 , 20
7 , . . . } set of rational numbers
R = {0, 1, 3.1, , e, . . . } set of real numbers
C = {1 + i1, 1 + i3.4, . . . } set of complex numbers
Remark 3. All above sets apart from have infinite number of elements.
Properly Formed Sets
Russells paradox (after Bertrand Russell, 18721970)
A barber in a village put a note:
I shave everybody who does not shave themselves
Question 1. Does the barber shave himself ?
The following sentence is false The previous sentence is true
imR = {b B : (a, b) R}
imR = {0, 1}
= {(0, ), (1, )}
3
Functions (Mappings)
Definition 10 (Function (mapping)). A function (mapping) from X into Y
f :XY ,
or y = f (x)