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Math 3121 Abstract Algebra I. Lecture 2 Sections 0-1: Sets and Complex Numbers. Questions on HW (not to be handed in). HW: pages 8-10: 12, 16, 19, 25, 29, 30. Finish Section 0: Sets and Relations. Correction to slide on functions Equivalence Relations and Partitions.

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math 3121 abstract algebra i

Math 3121Abstract Algebra I

Lecture 2

Sections 0-1: Sets and Complex Numbers

questions on hw not to be handed in
Questions on HW (not to be handed in)
  • HW: pages 8-10: 12, 16, 19, 25, 29, 30
finish section 0 sets and relations
Finish Section 0: Sets and Relations
  • Correction to slide on functions
  • Equivalence Relations and Partitions
corrected slide functions
Corrected Slide: Functions
  • Definition: A function f mapping a set X into a set Y is a relation between X and Y with the properties:

1) For each x in X, there is a y in Y such that (x, y) is in f

2) (x, y1) ∊ f and (x, y2) ∊ f implies that y1 = y2.

  • When f is a function from X to Y, we write

f: X Y, and we write “(x, y) in f” as “f(x) = y”.

functions corrected version
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.
  • This means that

1) For each x in X, there is a y in Y such that (x, y) is in f

2) (x, y1) ∊ f and (x, y2) ∊ f implies that y1 = y2.

  • When f is a function, we write “(x, y) in f” as “f(x) = y”.
recall equivalence relation
Recall: Equivalence Relation
  • Definition: An equivalence relation Ron a set S is a relation on S that satisfies the following properties for all x, y, z in S.
    • Reflexive: x R x
    • Symmetric: If x R y, then y R x.
    • Transitive: If x R y and y R z, then x R z.
equivalence classes
Equivalence Classes
  • Definition: Suppose ~ is an equivalence relation on a nonempty set S. For each a in S, let a̅ = {x∊S | x~a}. This is called the equivalence class of a ∊ S with respect to ~.
notes about equivalence
Notes about Equivalence

1) By definition of a̅ = {x ∊ S | x ~ a}:

x ∊ a̅ ⇔ x ~ a.

2) By symmetry, a ~ a. Thus:

a ∊ a̅

functions and equivalence relations
Functions and Equivalence Relations

Theorem: Suppose f: X  Y is a function from a set X to a set Y. Define a relation ~ by (x ~ y)⇔ (f(x) = f(y)). Then ~ is an equivalence relation on X.

Proof: Left to the reader.

theorem
Theorem
  • Theorem: Let ~ be an equivalence relation on a set S, and let a̅ denote the equivalence class of a with respect to ~. Then

x ~ y ⇔ x̅ = y̅.

proof
Proof

Proof: We show each direction of the implication separately

x ~ y ⇒ x̅ = y̅:

We will show that x ~ y implies that x̅ and y̅ have the same elements.

1) Start with transitivity of ~: x ~ y and y ~ z ⇒ x ~ z

2) Rewrite 1) as: x ~ y ⇒ (y ~ z ⇒ x ~ z) (Note: P and Q ⇒ R is equivalent to P ⇒ (Q ⇒ R ))

3) By symmetry of ~, replace y ~ z by z ~ y and x ~ z by z ~ x in 2):

x ~ y ⇒ (z ~ y ⇒ z ~ x)

4) Reversing x and y in 3) gives: y ~ x ⇒ (z ~ x ⇒ z ~ y).

5) By symmetry of ~, replace y ~ x by x ~ y in 4):

x ~ y ⇒ (z ~ x ⇒ z ~ y).

6) Combining 5) and 3):

x ~ y ⇒ (z ~ x ⇔ z ~ y).

7) By definition of equivalence class

x ~ y ⇒ (z ∊ x̅ ⇔ z ∊ y̅).

Thus x ~ y ⇒ x̅ = y̅

x̅ = y̅ ⇒ x ~ y :

Suppose x̅ = y̅. Since x ∊ x̅, equality of sets implies that x ∊ y̅. Thus x ~ y.

Thus x ~ y ⇔x̅ = y̅.

QED

partitions
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.
  • Note that a subset P of the power set of S is a partition whenever

1) ∀ x ∊ S, x is in some member of P

2) ∀ X, Y ∊ P, (X⋂Y ≠ Ø) ⇒ X=Y.

  • Note that 1) is equivalent to:

1’) The union of all members of P is equal to S:

      • ∪(P) = S

and 2) is equivalent to:

2’) The intersection of any two different members of P is empty.

      • ∀ X, Y ∊ P, X ≠ Y ⇒X ÇY =Ø
theorem13
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 whose members are the equivalence classes a̅ = {x ∊ S | x ~ a}.
proof14
Proof

Proof: Let P = {a̅ | a ∊ S}

We show that P is a partition of S. We must show that P is a collection of nonempty subsets of S such that each element of S is in exactly one member of P.

We will show that 1) for each x in S, x ∊ x̅, 2) for any x,y in S, (x̅⋂y̅ ≠ Ø) ⇒ (x̅=y̅). From 1) we conclude that each member of P is nonempty, and that each member of S is in at least one member of P. From 2) we conclude that each element of S is in at most one member of P.

proof of 1
Proof of 1)

Proof of 1) For each a ∊ S, a ∊ a̅:

Let a ∊ S.

Because ~ is reflexive, a ~ a

Thus a ∊ a̅ = {x ∊ S |x~a} .

proof of 2
Proof of 2)

Proof of 2) for any x,y in S, (x̅⋂y̅ ≠ Ø)⇒ (x̅=y̅):

Assume x̅⋂y̅ ≠ Ø. Thenthere is an element z in x̅ ⋂ y̅. Then z ~ x and z ~ y. Applying symmetry to the first and then transitivity to the pair, we get x ~ y. By the previous theorem x̅ = y̅.

Thus (x̅⋂y̅ ≠ Ø)⇒ (x̅=y̅).

section 1 complex numbers
Section 1: Complex Numbers
  • This section covers complex numbers. It summarizes:
    • Definition: ℂ = { a + b i | a, b ∊ ℝ}, where i2 = -1
    • Addition and multiplication of complex numbers:
      • (a + b i)+(c + d i) = (a + c) + (b + d) i
      • (a + b i)(c + d i) = a c + a d i + b i c + b i d i

= (a c – bd) + (a d + b c) i

Note: follows from distributive and commutative laws.

    • Absolute value: |a + b i | = sqrt (a2 + b2)
    • Euler’s formula: e i ϑ = cos ϑ + i sin ϑ.
    • Polar coordinates in the complex plane.

r e i ϑ =r cos ϑ + i r sin ϑ.

    • Solving for roots using polar coordinates.
    • The unit circle in the complex plane.
    • Roots of unity: e i 2π/n = cos (2π/n) + i sin (2π/n)
more on the complex unit circle
More on the Complex Unit Circle
  • Let U is the unit circle on the complex plane.

U = {z ∊ ℂ | |z| = 1}

  • U is closed under multiplication of complex numbers.
  • The function f(ϑ) = e i ϑmaps the real numbers into the complex circle. It wraps the real line around the circle.
  • Note the addition formula: f(a+b) = f(a) f(b). Expand this in terms of Euler and get the addition formulas for sine and cosine (in class).
  • Note that a ~ b ⇔ f(a) = f(b) defines an equivalence relation on ℝ.
hw not to hand in
HW – not to hand in
  • Pages 19-20: 1, 3, 5, 13, 17, 23, 38, 41