The document discusses integers and their properties. It defines integers as numbers that can be written without fractions or decimals, such as 21, 4, and -2048, but not 9.75 or √2. Integers include natural numbers, zero, and their negatives. They form the smallest group containing natural numbers under addition. Integers also form a ring with unique homomorphisms to other rings, characterizing their fundamental nature.

1. The basic concept of sets, its elements and
subsets.
Concept of Sets
The word "set" implies a collection or grouping of similar objects or symbols. The objects in
a set have at lea& one characteristic in common, such as similarity of appearance or
purpose. A set of tools would be an example of a group of objects not necessarily similar in
appearance but similar in purpose. The objects or symbols in a set are called members or
Element of Set
The elements of a mathematical ret are usually symbols, ouch as numerals , lines, or points.
For example, the integer6 greater than zero and less than 6 form a set, as follows:
{1, 2, 3, 4}
Notice that braces are used to indicate sets. This is often done where the elements of the set
are not too numerous, Since the elements of the set (2, 4, 6) are the same as the elements of
(4, 2, 6}, there two seta are said to be equal. In other words, equality between sets has
nothing to do with the order in which the elements are arranged. Further- more, repeated

2. elements are not necessary. That is, the elements of (2, 2, 3, 4) are simply 2, 3, and 4.
Therefore the sets (2, 3, 4) and (2, 2, 3,4) are equal. Practice problems:
1. Use the correct symbols to designate the set of odd positive integers greater than 0 and
less than 10.
of names of days of the week which do not contain the letter "s".
3. List the elements of the set of natural numbers greater than 15 and less than 20.
4. Suppose that we have sets as follows:
A = (1, 2, 3) C = (1, 2, 3, 4)
B = (1, 2, 2, 3) D = (1, 1, 2, 3)
Which of these sets are equal?
Answers:
1. 1, 3, 5, 7, 9
2. {Monday, Friday}
3. 16, 17, 18, and 19

3. 4. A=B=D
SUBSETS
Since it is inconvenient to enumerate all of the elements of a set each time the set is
mentioned, sets are often designated by a letter. For example, we could let S represent the
set of all integers greater than 0 and less than 10. In symbols, this relationship could be
stated as follows:
s = (1, 2, 3, 4, 5, 6, 7, 8,9)
Now suppose that we have another set, T, which comprises all positive even integers less
than 10. This set is then defined as follows:
T - (2, 4, 6, 8)
Notice that every element of T is also an element of S. This establishes
the SUBSET relationship; T is said to be a subset of 9.

4. A Venn diagram or set diagram is a diagram that shows all possible logical relations between a finite
collection of sets. Venn diagrams were conceived around 1880 by John Venn. They are used to teach
elementary set theory, as well as illustrate simple set relationships
in probability, logic,statistics, linguistics and computer science.
Sets A (creatures with two legs) and B (creatures that can fly)
This example involves two sets, A and B, represented here as coloured circles. The orange circle, set A,
represents all living creatures that are two-legged. The blue circle, set B, represents the living creatures
that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram. Living
creatures that both can fly and have two legs—for example, parrots—are then in both sets, so they
correspond to points in the area where the blue and orange circles overlap. That area contains all such
and only such living creatures.
Humans and penguins are bipedal, and so are then in the orange circle, but since they cannot fly they
appear in the left part of the orange circle, where it does not overlap with the blue circle. Mosquitoes have
six legs, and fly, so the point for mosquitoes is in the part of the blue circle that does not overlap with the
orange one. Creatures that are not two-legged and cannot fly (for example, whales and spiders) would all
be represented by points outside both circles.
The combined area of sets A and B is called the union of A and B, denoted by A∪ B. The union in this
case contains all living creatures that are either two-legged or that can fly (or both).
The area in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A
∩ B. For example, the intersection of the two sets is not empty, because there arepoints that represent
creatures that are in both the orange and blue circles.

5. Set B is a subset of a set A if and only if every object of B is also an object of A.
We write B A
By definition, the empty set( { } or ) is a subset of every set
Now, take a look at the following Venn diagrams.
Definition of Venn Diagrams:
Venn Diagrams are closed circles, named after English logician Robert Venn, used to represent
relationships between sets
B = { a, b, c}
A = { a, b, c, f}
U = { a, b, c, f}
Since all elements of B belong to A, B is a subset of A
Proper subset:
Set B is a proper subset of set A, if there exists an element in A that does not belong to B.
we write B A
Having said that, B is a proper subset of A because f is in A, but not in B.
We write B A instead of B A
Universal set:

6. The set that contains all elements being discussed
In our example, U, made with a big rectangle, is the universal set
Set A is not a proper subset of U because all elements of U are in subset A
Notice that B can still be a subset of A even if the circle used to represent set B was not inside the
circle used to represent A. This is illustrated below:
As you can see, B is still a subset of A because all its objects or elements (c, and d) are also objects
or elements of A.
B is again a proper subset because there are elements of A that does not belong to B
A and B are also subsets of the universal set U, but especially proper subsets since there are
elements in U that does not belong to A and B
In general, it is better to represent the figure above as show below to avoid being redundant:
The area where elements c, and d are located is the intersection of A and B. More on this on a
different lesson!
If you have any questions about the subset of a set, I will be more than happy to answer them.

7. In this lesson we look at some properties that apply to all real numbers. If you learn these
properties, they will help you solve problems in algebra. Let's look at each property in
detail, and apply it to an algebraic expression.
#1. Commutative properties
The commutative property of addition says that we can add numbers in any order. The
commutative property of multiplication is very similar. It says that we can multiply numbers
in any order we want without changing the result.
addition
5a + 4 = 4 + 5a
multiplication
3 x 8 x 5b = 5b x 3 x 8
#2. Associative properties
Both addition and multiplication can actually be done with two numbers at a time. So if
there are more numbers in the expression, how do we decide which two to "associate"
first? The associative property of addition tells us that we can group numbers in a sum in
any way we want and still get the same answer. The associative property of multiplication
tells us that we can group numbers in a product in any way we want and still get the same
answer.
addition
(4x + 2x) + 7x = 4x + (2x + 7x)
multiplication
2x2
(3y) = 3y(2x2
)
#3. Distributive property
The distributive property comes into play when an expression involves both addition and
multiplication. A longer name for it is, "the distributive property of multiplication over
addition." It tells us that if a term is multiplied by terms in parenthesis, we need to
"distribute" the multiplication over all the terms inside.
2x(5 + y) = 10x + 2xy
Even though order of operations says that you must add the terms inside the parenthesis first, the distributive
property allows you to simplify the expression by multiplying every term inside the parenthesis by the multiplier.
This simplifies the expression.
#4. Density property
The density property tells us that we can always find another real number that lies between
any two real numbers. For example, between 5.61 and 5.62, there is 5.611, 5.612, 5.613
and so forth.
Between 5.612 and 5.613, there is 5.6121, 5.6122 ... and an endless list of other numbers!

8. #5. Identity property
The identity property for addition tells us that zero added to any number is the number
itself. Zero is called the "additive identity." The identity property for multiplication tells us
that the number 1 multiplied times any number gives the number itself. The number 1 is
called the "multiplicative identity."
Addition
5y + 0 = 5y
Multiplication
2c × 1 = 2c

9. An integer is a number that can be written without a fractional or decimal component. For example, 21, 4,
and −2048 are integers; 9.75, 5½, and √2 are not integers. The set of integers is a subset of the real
numbers, and consists of the natural numbers (1, 2, 3, ...), zero (0) and the negatives of the natural
numbers (−1, −2, −3, ...).
The name derives from the Latin integer (meaning literally "untouched," hence "whole": the
word entire comes from the same origin, but via French
[1]
). The set of all integers is often denoted by a
boldface Z (or blackboard bold , Unicode U+2124 ℤ), which stands for Zahlen (German for numbers,
pronounced [ˈtsaˈlən]).
[2][3]
The integers (with addition as operation) form the smallest group containing the additive monoid of the
natural numbers. Like the natural numbers, the integers form acountably infinite set. In algebraic number
theory, these commonly understood integers, embedded in the field of rational numbers, are referred to
as rational integers to distinguish them from the more broadly defined algebraic integers.
The integers (with addition and multiplication addition) form a unital ring which is the most basic one, in
the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this
ring. This universal property characterizes the integers.