Math 3121Abstract Algebra I

Section 0: Sets

The axiomatic approach to Mathematics

• The notion of definition - from the text: "It is impossible to define every concept.“

• Why?– Rather: concepts are presented as systems which have

components, operations, relations, and constraints.– The system is defined as a whole.– This is the approach that is used in modern

mathematics, including Abstract Algebra.

Examples of Axiomatic Systems

• Euclidean Geometry– components: point, straight line, circle, angle,

plane– relations: lies on, in between, meets, congruent– constraints: axioms (also called postulates)

Euclid’s Axioms

• Euclid's postulates from Coxeter's Geometry, page 4:– 1) A straight line may be drawn from any point to any other point.– 2) A finite straight line may extended continuously in a straight line.– 3) A circle may be described with any center and any radius.– 4) All right angles are equal to one another.– 5) If a straight line meets two other straight lines so as to make the

two interior angles on one side of it together less than two right angles, the straight lines, if extended indefinitely, will meet on that side on which the angles are less than two right angles.

• Note: Hilbert improved upon these with his list of twenty. Can you find these on line?

Examples of Axiomatic Systems

• Set Theory– components: elements, sets– relations: belongs to, is included in– operations: union, intersection, complement– constraints: axioms

Foundations

• Abstract Algebra takes an axiomatic approach and is built on the foundation of Set Theory.

• Set theory is built on a foundation of logic.• However, there are several versions of set

theory and several versions of logic.

Formal and Informal Approaches• Natural languages (such as Greek, Latin, Arabic, Chinese, or

English) have been used to describe mathematics.• However, natural language is subject to interpretation and is often

ambiguous.• It is not natural to use a lot of punctuation (such as parentheses)

to group words so that meaning is clear.• Formal language was developed to make communication more

precise.• However, humans have a hard time with formalities, but rely on

them just the same. • Both formal and informal language is essential for mathematical

understanding and creativity.

Summary of (formal classical) Logic• Propositional logic:

– components: propositions (well-formed formulas = wff)• if P and Q are wff, then P → Q and ¬ P are also.

– relations: true, false– primitive operations: implies, not, modus ponens

• implies: →• not: ¬• modus ponens; if P and P → Q, then Q.

– defined operations: and, or– axioms (many possibilities). For example: start with saying that the following are true:

• L1: (A →(B → C))• L2: ((A →(B →)) →((A → B) →(A → C))• L3: (((¬ B) →(¬ A)) →(((¬ B) → A) → B))

– Note: can use truth tables to verify truth. (example in class)• Predicate calculus:

– add quantifiers: (for all) and (there exists)– Truth is not as easy as with propositional logic.

Summary of Set Theory• ZFC = Zermelo, Frankel, with Axiom of choice

– components: sets– primitive relations: belongs (in), equals (=)– defined operations: union, intersection, singleton formation (x -> {x}).– axioms (informally stated by Halmos & Suppes - also Goldblatt):

• 1) Extension: Sets A and B are equal and only if they have the same elements.• 2) Specification: To every set A and to every condition S(x) there is a set B whose elements are precisely those elements x of A

for which S(x) holds.• 3) Pairing: For any two sets there exists a set that they both belong to.• 4) Unions: For every collection of sets there exists a set that contains all the elements that belong to at least one set of the given

collection.• 5) Powers: For each set there is a set to which all subsets of A belong.• 6) Regularity: For any nonempty set A, there is an element of A that is disjoint from A.• 7) Infinity: There is a set that contains the empty set and is closed under the operation• x |-> x union {x}• 8) Replacement: If the domain of a function is a set, then so is its range.• 9) Choice: (many forms - later as time permits)

– Notes: • Replacement => Specification• Replacement + Powers => Pairing

• NBG = Von Neumann, Bernays, Gödel– components: sets and classes

Paradoxes (why so many axioms)

• Russell: The class of all sets that do not belong to themselves is not a set.

• Curry: If this sentence is true, then Santa Claus exists.

Working knowledge of set theory• definitions and examples of• empty set (page 1)• sets of numbers (natural, integer, rational, real, complex, quaternion)• set builder notation {}• subset (page 2)• proper and improper subset (page 2)• ordered pair (page 3)• Cartesian product (page 3)• relation between sets (page 3)• function (page 4)• one-to-one functions (page 4)• onto functions (page 4)• inverse function (page 5)• cardinality (page 5)• partition (page 6)• equivalence relation (page 7)• reflexive, symmetric, transitive• theorem: equivalence relations and partitions (and functions)

Set Builder Notations

• Bracketed lists separated by comma:For example {1, 2, 5}

• Specification by a property P(x):{x | P(x)}Note: No guarantee that this is a actually a set.Usually write instead:{ x in A | P(x) }

• Examples in class.

Familiar Sets of Numbers

• Natural Numbers: {1, 2, 3, …}

• Integers (Whole Numbers): positive, negative, and zero:

{…, -3, -2, -1, 0, 1, 2, 3,…}

• Rational Numbers: fractions of integers{m/n | m and n integers with n not zero}

• Real Numbers• Complex Numbers• And more – see book for notations

Cartesian Product

• Definition: The Cartesian product of two sets A and B is a set A×B consisting of all ordered pairs (a, b) with a in A and b in B.

Relation

• Definition: A relation between sets A and B is a subset R of their Cartesian product A×B. We read (a, b) in R as “a is related to b (via R)” and write “a R b”.

Functions (Corrected Version)

• Definition: A function f mapping a set X into a set Y is a relation between X and Y with the property that each x in X appears exactly once as the first element of an ordered pair (x, y) in f. In that case we write f: X Y.

• When f is a function, we write “(x, y) in f” as “f(x) = y”.

Domain and Codomain

• Let f: X Y be a function, thenThe Domain of f = XThe Codomain of f = YThe Range (or Image) of f = f[X] = {f(x) | x in X}

One-to-One and Onto Functions

• Definition: A function f: X Y is one-to-one iff f(x1)=f(x2) implies that x1 = x2.

• Definition: A function f : X Y is onto iff the range of f is the codomain Y.

Equivalence Relation

• Definition: An equivalence relation R on a set S is a relation on S that satisfies the following properties for all x, y, z in S.

.then, andIfe)(Transitiv.3.then,If)(Symmetric.2

.R)(Reflexive.1

xRzyRzxRyyRxxRy

xx

Partitions

• Definition: A partition of a set S is a set P of nonempty subsets of S such that every element of S is in exactly one of the subsets of P. The subsets (elements of P) are called cells.

• When discussing a partition of a set S, we denote by ū the cell containing the element u of S.

Theorem

• Theorem (Equivalence Relations and Partitions): Let S be a nonempty set and let ~ be an equivalence relation on S. Then ~ corresponds to a partition of S such that

}~{ axSxa

HW

• HW: pages 8-10: 12, 16, 19, 25, 29, 30