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Math 3121 Abstract Algebra I Lecture 2 Sections 0-1: Sets and Complex Numbers

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Questions on HW (not to be handed in) HW: pages 8-10: 12, 16, 19, 25, 29, 30

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Finish Section 0: Sets and Relations Correction to slide on functions Equivalence Relations and Partitions

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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, y 1 ) f and (x, y 2 ) f implies that y 1 = y 2. 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.

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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, y 1 ) f and (x, y 2 ) f implies that y 1 = y 2. When f is a function, we write (x, y) in f as f(x) = y.

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Recall: 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. 1.Reflexive: x R x 2.Symmetric: If x R y, then y R x. 3.Transitive: If x R y and y R z, then x R z.

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Equivalence Classes Definition: Suppose ~ is an equivalence relation on a nonempty set S. For each a in S, let a̅ = {xS | x~a}. This is called the equivalence class of a S with respect to ~.

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Notes about Equivalence 1) By definition of a̅ = {x S | x ~ a}: x a̅ x ~ a. 2) By symmetry, a ~ a. Thus: a a̅

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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.

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Theorem Theorem: Let ~ be an equivalence relation on a set S, and let a ̅ denote the equivalence class of a with respect to ~. Then x ~ y x̅ = y̅.

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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

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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, ( XY 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 Y =Ø

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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}.

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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, ( x̅y̅ 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.

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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}.

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Proof of 2) Proof of 2) for any x, y in S, ( x̅y̅ Ø) (x̅=y̅): Assume x̅y̅ Ø. Then there 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 ( x̅y̅ Ø) (x̅=y̅).

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Section 1: Complex Numbers This section covers complex numbers. It summarizes: – Definition: = { a + b i | a, b }, where i 2 = -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 (a 2 + b 2 ) – Eulers 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)

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More on the Complex Unit Circle Let U is the unit circle on the complex plane. U = {z | |z| = 1} U is closed under multiplication of complex numbers. The function f( ϑ) = e i ϑ maps 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.

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HW – not to hand in Pages 19-20: 1, 3, 5, 13, 17, 23, 38, 41

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