Algebraic Structures And The Algebra Of Numbers

Algebraic Structures and the Algebra of Numbers

Ways of Describing Sets

A SET is a collection of objects (empty or nonempty) subject to the following conditions: - A collection must be well-defined (there is no ambiguity whether an object does or does not belong to the set) - Each unique object should be uniquely represented - Order of representing each object is immaterial

1. Roster Method (Enumeration, Listing Method)

Sets are usually denoted by capital letters, and objects of the set (called elements) are usually denoted by small letters. We use { } to enclose the elements of the set and each object is separated by a comma. Example: Consider A as the collection of vowels in the English alphabet A = { a, e, i, o, u} We use the symbol ∈ to denote that an object belongs to the set. Otherwise, we use ∉ . For our example, a∈ A o∈ A

Example: Let C be the collection of primary colors (C is a set). C = { blue, yellow, red } 2. Rule Method (Set Builder Notation) – a descriptive phrase is used to describe the elements of the set

{ x x is a _______ } or { x ∈ A x is a _______ } Example: Let F be the collection of the names (initials) of all colleges in UPLB at present. D = { x x is a name of college in UPLB at present} Set Relations

b∉ A

k∉A

A set is said to be finite if the number of elements in the set is zero or a counting number. An empty set is a set with no elements. It is considered as a finite set and is denoted by ∅ or { } . The cardinality of a set, say S, is the total number of elements on the set. It is denoted by n( S ) . The universal set, denoted by U, is the set of all elements under consideration.

A set A is said to be a subset of set B if all elements of A is in B. A ⊆ B ⇔ for all x ∈ A then x ∈ B Two sets A and B are said to be equal if A is a subset of B and B is a subset of A. They are the same sets, which are just labeled/ represented/ described differently. A = B ⇔ A ⊆ B and B ⊆ A A nonempty set A is said to be a proper subset of set B if A is a subset of B but is not equal to B. A ⊂ B ⇔ for A ≠ ∅, A ⊆ B and A ≠ B

Note: Every nonempty set has two improper subsets, the empty set and the set itself.

Remark: If A ∩ B = ∅ , then set A and B are said to be disjoint.

A one-to-one correspondence is said to exist between sets A and B if it is possible to associate the elements of A to elements of B in such a way that each element of B is associated with exactly one element of A. 1−1 A → B

The Set of Real Numbers

Two sets A and B are equivalent if there exists a one-to-one correspondence between them. 1−1 A ~ B ⇔ there is A → B

Note: If n( A) = n( B ) for finite sets A and B then A ~ B .

Natural Numbers:

= {1, 2, 3, 4, 5, 6, }

Whole Numbers:

= { 0,1, 2, 3, 4, 5, 6, } = ∪ { 0}

Integers: Subsets of the Set of Integers

∪ {− n n ∈ }

=

(set of positive integers) (set of negative integers)

A set is said to be infinite if it is equivalent to one of its proper subsets. A is infinite ⇔ Given B ⊂ A and A ~ B .

(set of non-negative integers)

Note: Let I be an infinite set. Then n( I ) = ℵ ( ∞ ).

(set of non-positive integers) = { 0,1, 2, 3,  , n − 1} where n ≥ 2

Set Operations Let A and B be subsets of some universal set U. Set Complement:

A' = A c = { x x ∈ U ∧ x ∉ A}

Union:

A ∪ B = { x x ∈ A ∨ x ∈ B}

Intersection:

A ∩ B = { x x ∈ A ∧ x ∈ B}

Set Difference:

A − B = A \ B = { x x ∈ A ∧ x ∉ B}

Cross Product:

A × B = {( a, b ) x ∈ A ∧ x ∈ B}

= {nk n ∈ Rational Numbers:

}

p  =  p, q ∈ Z , q ≠ 0 q 

Irrational Numbers: (set of non-terminating, non-repeating decimals) Real Numbers:

=

Complex Numbers:

= {a + bi a, b ∈

The Real Number Line

, i 2 = −1}

Note: There is a one-to-one correspondence of the set of real numbers and the set of points on a line.

There are six basic operations on the set of real numbers. Some Properties:

Order: The set of real numbers can be ordered in such a way that given a set of distinct real numbers, there will always be a least and a greatest value. In other words, all real numbers can be arranged by way of less than and greater than. Completeness: Given distinct real numbers, there will always be another real number between the two. This is much the same as the set of points on the real number line where there will always exist another point between two distinct points. Interval Notations Bounded Intervals Open Interval: Closed Interval Some Variations [ a, b ) = { x ∈

( a, b ) = {x ∈ [ a, b ] = { x ∈ a ≤ x < b}

a < x < b} a ≤ x ≤ b}

( a, b ] = {x ∈

a < x ≤ b}

Unbounded Intervals

( a,+∞ ) = {x ∈ ( − ∞, b ) = { x ∈

x > a} x < b}

[ a,+∞ ) = {x ∈ ( − ∞, b ] = { x ∈

x ≥ a} x ≤ b}

is closed under the operations + and × For any a, b ∈ , a + b ∈ and a × b ∈ . + and × are commutative on For any a, b ∈ , a + b = b + a and a × b = b × a + and × are associative on For any a, b ∈ , ( a + b ) + c = a + ( b + c ) and ( a × b) × c = a × ( b × c ) × is distributive over + For any a, b, c ∈ , a × ( b + c ) = ( a × b ) + ( a × c ) and ( a + b) × c = ( a × c ) + ( b × c )

Definitions: Let m, n ∈

.

a 0 = 1 , where a ≠ 0 1 a − n = n , where a ≠ 0 a a a a

Some Operations on the Real Numbers

, - and ÷ are neither commutative nor associative.

Note: In

1

n

m

=n a =

n

−m

n

( a)

=

n

m

= n a m provided GCD{ m, n} = 1

1 m

a n Properties of Exponents. Given a, b ∈

and m, n ∈ .

1. a m ⋅ a n = a m + n

4. ( ab ) n = a n ⋅ b n

am 2. n = a m −n , a ≠ 0 a

an a 5.   = n , b ≠ 0 b b

( )

3. a m

n

n

Let G be a nonempty set and ∗ be a binary operation on G. The structure G,∗ is said to be a group if the following conditions are satisfied: i. ∗ is associative on G. For any a, b, c ∈ G, a ∗ ( b ∗ c ) = ( a ∗ b ) ∗ c

= a m⋅n

ii. There exists an identity element, say e. e ∈ G such that a ∗ e = e ∗ a = a for any a ∈ G

Properties of Radicals. 1.

2.

n

a ⋅ b = a ⋅b

n

a

n

n

b

=n

n

a b

3.

m n

4.

n

a =

mn

a

a if n is odd an =   a if n is even

The Algebraic Structures An algebraic structure is just a nonempty set and some operations on this set. Consider the notation A,∗ for a nonempty set A and an operation ∗ on the elements of A, or operation on A.

A,∗, where  is another

Binary Operations Consider a nonempty set A. A binary operation on A is a rule that assigns to each element of A × A is a unique element of A. An operation ∗ on a set A is said to be a binary operation on A if a ∗ b ∈ A and a ∗ b is unique for any a, b ∈ A . Groups

iii. Each element a has a corresponding inverse element, say a ' . If a ∈ G , there should exists a '∈ G such that a ∗ a' = a' ∗ a = e Fields Let F be a nonempty set and +, × be binary operations on F. The structure F ,+,× is said to be a field if the following conditions are satisfied: i. + and × are commutative on F. For any a, b ∈ F , a + b = b + a and a × b = b × a ii. + and × are associative on F. For any a, b, c ∈ F , a + ( b + c ) = ( a + b ) + c a × ( b × c) = ( a × b) × c iii. Existence of identity elements for + and × . There exist 0 ∈ F (the additive identity) such that a + 0 = 0 + a = a for any a ∈ F There exists 1 ∈ F (the multiplicative identity) such that a × 1 = 1 × a = a for any a ∈ F iv. Existence of an additive inverse for each element and a multiplicative inverse for each non-zero element.

For

each a ∈ F , there exists a + ( − a) = ( − a) + a = 0

− a ∈ F such

that

: Given a, b, c ∈ ,

Equality Axioms in

For each a ∈ F , a ≠ 0 , there exists a −1 ∈ F such that a × a −1 = a −1 × a = 1 v. × is distributive over + For any a, b, c ∈ F , a × ( b + c ) = ( a × b ) + ( a × c ) ( a + b) × c = ( a × c ) + ( b × c )

i. Reflexivity:

a=a

ii. Symmetry:

If a = b , then b = a

iii. Transitivity:

If a = b and b = c , then a = c

iv. Addition Property of Equality: If a = b , then a + c = b + c and c + a = c + b

Algebraic Expressions

v. Multiplication Property of Equality: If a = b , then a × c = b × c and c × a = c × b

Operations on Polynomials Factoring Polynomials

Inequalities

Rational Expressions

Order Axioms in

Definition: Given a polynomial P( x ) . A is a zero of P( x ) if P( a ) = 0

ii. Transitivity:

Theorem: If x − a is a factor of P( x ) , then a is a zero of P( x ) . The Remainder Theorem: The remainder when P( x ) is divided by x − a is P( a ) Theorem: The zeros of the polynomial

ax 2 + bx + c, a ≠ 0

are

− b + b − 4ac − b − b − 4ac and . 2a 2a The Fundamental Theorem of Algebra: Every polynomial of degree n has exactly n zeros (up to multiplicity). 2

: Given a, b, c ∈ ,

i. Trichotomy Axiom: If a, b ∈ , only one of the statements hold: ab a=b

Zeros of Polynomials

2

Equations

If a < b and b < c , then a < c .

iii. Addition Property of Inequality: If a < b , then a + c < b + c iv. Multiplication Property of Inequality: If a < b and c > 0 , then a × c < b × c . If a < b and c < 0 , then a × c > b × c