Garis-garis Besar Perkuliahan

Garis-garis Besar Perkuliahan 15/2/10 Sets and Relations 22/2/10 Definitions and Examples of Groups 01/2/10 Subgroups 08/3/10 Lagrange’s Theorem 15/3/10 Mid-test 1 22/3/10 Homomorphisms and Normal Subgroups 1 29/3/10 Homomorphisms and Normal Subgroups 2 05/4/10 Factor Groups 1 12/4/10 Factor Groups 2 19/4/10 Mid-test 2 26/4/10 Cauchy’s Theorem 1 03/5/10 Cauchy’s Theorem 2 10/5/10 The Symmetric Group 1 17/5/10 The Symmetric Group 2 22/5/10 Final-exam

Sets and Relations Section 0

Sets • A 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 setA, 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 • Show that a set having n elements has 2n subsets. • 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 onS.

Functions • A functionf : 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 functionf : X  Y and a subset B Y, we define • f -1(B) is called the inverse image of Bunderf. • The inverse image of the subset {y} of Y is simply denoted by f -1(y).

One-to-one and Onto Functions • A 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 orsurjective if the range of f is Y. • The function f : X  Y is said to be a 1-1correspondence 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.

Problems • If is onto and are such that g∘f = h∘f, prove that g = h. • 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,

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 relationR 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. • S = R reals, a ~ b if a – b is rational. • S = straight lines in the plane, a ~ b if a, b are parallel. • 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 R is a binary operation. • So is the usual multiplication on R. • We could just as well replace R with R+, C, Z, or Z+ in the previous examples. • Matrix addition on M2x2(R) – 2x2 matrices, is a binary operation. • Matrix addition on M(R) – 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

Problems Let S consist of the two objects  and . We define the operation  on S by subjecting  and  to the following conditions: •    =  =    •    =  •    = 

Problems (cont.) Verify by explicit calculation that • S is closed under . •  is commutative •  is associative • There is a particular e (identity element) in S such that eb = be = b for all b in S • Given b in S, then bb = e, where e is the particular element in Part (4).

Problems Each 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 ^)=

Garis-garis Besar Perkuliahan