Garis-garis Besar PerkuliahanGaris-garis Besar Perkuliahan15/2/10 Sets and Relations22/2/10 Definitions and Examples of Groups01/2/10 Subgroups08/3/10 Lagrange’s Theorem15/3/10 Mid-test 122/3/10 Homomorphisms and Normal Subgroups 129/3/10 Homomorphisms and Normal Subgroups 205/4/10 Factor Groups 112/4/10 Factor Groups 219/4/10 Mid-test 226/4/10 Cauchy’s Theorem 103/5/10 Cauchy’s Theorem 210/5/10 The Symmetric Group 117/5/10 The Symmetric Group 222/5/10 Final-exam

Sets and RelationsSection 0

SetsA set S is made up of elements. a S

means that a is an element of S.There is exactly one set with no elements,

the empty set, .Sets are described by

◦ Listing the elements, or◦ Giving a characterizing property of its elements

A set is well defined – given a set S and an object a, either a is definitely an element of S or it definitely is not an element of S.

Subsets

• A set B is a subset of a set A, B A, if every element of B is an element of A.

• B A means B A but B A• If A is any set, then A is an improper subset of A.

Any other subset of A is a proper subset of A• Given sets A and B, the Cartesian product of A

and B is A B = {(a, b)| a A and b B}

Problems

1. Show that a set having n elements has 2n subsets.

2. If 0 < m < n, how many subsets are there that have exactly m elements?

Relations

• A relation between sets A and B is a

subset R of A B. We read (a, b) R

as “a is related to b,” and write aRb.

• A relation between a set S and itself will

be referred to as a relation on S.

FunctionsA function f : X Y is a relation between X

and Y such that each x X appears in exactly one ordered pair (x, y) in f.

f is also called a map or mapping of X into Y. We express (x, y) f as f(x) = y.The domain of f is X and the codomain of f

is Y.The range of f is f(X)={f (x) | x X}.

Inverse Image

Given a function f : X Y and a subset B Y,

we define

f -1(B) is called the inverse image of B under f.

The inverse image of the subset {y} of Y is

simply denoted by f -1(y).

1f B x X f x B

One-to-one and Onto FunctionsA function f : X Y is one-to-one (written 1-1)

or injective if f(x1) = f (x2) only when x1 = x2.A function f : X Y is onto or surjective if the

range of f is Y.The function f : X Y is said to be a 1-1

correspondence or bijective if f is both 1-1 and onto. It has an inverse function f -1 : Y X defined by the property that f -1(y) = x if and only if f (x) = y.

Composition of Functions• Given f : X Y and g : Y Z, we define the

composition (or product), denoted by g∘f, to be the function g∘f : X Z defined by (g∘f)(x) = g(f(x)) for every x X.

• If f : X Y, g : Y Z, and h : Z U, then

h (∘ g∘f) = (h ∘g)∘f .• If f : X Y and g : Y Z are both 1-1, then g∘f : X Z

is also 1-1.

• If f : X Y and g : Y Z are both onto, then g∘f : X Z is also onto.

Problems1. If is onto and are such that g∘f = h∘f,

prove that g = h.2. If is 1-1 and are such that f∘g = f∘h,

prove that g = h• If S is a finite set and f is a 1-1 mapping

of S, show that for some integer n > 0,

:f S T g T U: ,

:h S T:g S T

:h T U

:f T U

Partitions• A partition of a set S is a collection of

nonempty subsets of S such that every

element of S is in exactly one of the

subsets

• The subsets are called the cells of the

partition

Equivalence Relations• An equivalence relation R on a set S

is one that satisfies these three properties for all x,y,z S:– (reflexive) xRx

– (symmetric) if xRy then yRx

– (transitive) if xRy and yRz then xRz

Equivalence Relations & Partitions Theorem Let S be a nonempty set and let

~ be an equivalence relation on S. Then ~ yields a partition of S, where

[a] = {x S | x ~ a} form the cells.

Also, each partition of S gives rise to an equivalence relation ~ on S, where a ~ b iff a and b are in the same cell of the partition.

Problems Show that the relation ~ defined in the

previous remark is an equivalence relation.

Verify that the relation ~ is an equivalence relation on the set given.

a) S = reals, a ~ b if a – b is rational.b) S = straight lines in the plane, a ~ b if a, b

are parallel.c) S = set of all people, a ~ b if they have the

same color eyes.

Binary Operation• A binary operation on a set S is a function

that maps S S into S.

• For each (a, b) S S, we will denote the

element ((a, b)) as a b.

Examples• The usual addition + on the set is a binary

operation.• So is the usual multiplication on .• We could just as well replace with +, , ,

or + in the previous examples.• Matrix addition on M2x2() – 2x2 matrices, is

a binary operation.• Matrix addition on M() – all matrices with

real entries, is NOT a binary operation.

Closure• Let be a binary operation on a set S, and let

H be a subset of S. The subset H is closed

under if for all a, b H we have a*b H.

• The binary operation on H given by restricting

to H is the induced operation of on H.

Commutativity and Associativity

• A binary operation on a set S is commutative

if a b = b a for all a, b S.

• A binary operation on a set S is associative if

(a b) c = a (b c) for all a, b, c S.

Tables

• For a finite set, a binary

operation on that set

can be defined in a

table

* a b ca b c bb a c bc c b a

Problems

Let S consist of the two objects and . We define the operation on S by subjecting and to the following conditions:

1. = = 2. = 3. =

Problems (cont.)Verify by explicit calculation that

1. S is closed under .2. is commutative3. is associative4. There is a particular e (identity element)

in S such that eb = be = b for all b in S5. Given b in S, then bb = e, where e is

the particular element in Part (4).

ProblemsEach of the following is an operation ¤ on R.

Indicate whether or not (i) it is commutative, (ii) it is associative, (iii) R has an identity element with respect to ¤, (iv) every x R has an inverse with respect to ¤:

• x ¤ y = x + 2y + 4• x ¤ y = |x + y|• x ¤ y = max{x, y} • x ¤ y = (xy)(x + y + 1)-1 on the set of positive

real numbers.

Question?

If you are confused like this kitty is, please ask questions =(^ y ^)=