Garisgaris Besar Perkuliahan 15210 22210 01210 08310 15310
Garis-garis Besar Perkuliahan 15/2/10 22/2/10 01/2/10 08/3/10 15/3/10 22/3/10 29/3/10 05/4/10 12/4/10 19/4/10 26/4/10 03/5/10 10/5/10 17/5/10 Sets and Relations Definitions and Examples of Groups Subgroups Lagrange’s Theorem Mid-test 1 Homomorphisms and Normal Subgroups 2 Factor Groups 1 Factor Groups 2 Mid-test 2 Cauchy’s Theorem 1 Cauchy’s Theorem 2 The Symmetric Group 1 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 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 2 n 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 a. Rb. • A relation between a set S and itself will be referred to as a relation on S.
Functions A 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).
One-to-one and Onto Functions A function f : X Y is one-to-one (written 1 -1) or injective if f(x 1) = f (x 2) only when x 1 = x 2. 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.
Problems 1. If that g = h. 2. If • is onto and are such that g∘f = h∘f, prove 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 relation R on a set S is one that satisfies these three properties for all x, y, z S: – (reflexive) x. Rx – (symmetric) if x. Ry then y. Rx – (transitive) if x. Ry and y. Rz then x. Rz
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 = R 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 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 M 2 x 2(R) – 2 x 2 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: 1. � = = � 2. � � = � 3. = �
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 5. Given b in S, then b b = e, where e is the particular element in Part (4). 1. 2. 3. 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 + 2 y + 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 ^)=
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