Garis-garis Besar Perkuliahan
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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 ^)=