Math 3000 - Lectures #13 and #14

Equivalence Relations

Reflexive, Symmetric and Transitive

Equivalence Classes

Partitions

Order Relations

Def: A relation R on a set A is antisymmetric if, for all x,y in A, if x R y and y R x then x = y.

Examples:

  1. The natural ordering on the set of real numbers
  2. For any set A, the subset relation defined on the power set P(A).
  3. Integer division on the set of natural numbers

Def: A relation R on a set A is a partial order (or partial ordering) for A if R is reflexive, antisymmetric and transitive. A set A with a partial order is called a partially ordered set, or poset.

The examples above are all examples of posets.

Def: Let R be a partial ordering on a set A and let a,bA with ab. Then a is an immediate predecessor of b if a R b and there does not exist cA such that ca, cb, a R c and c R b.

Def: A Hasse diagram for a partial order is a digraph representing this relation in which only the arcs to immediate predecessors are drawn and the digraph is drawn so that all arcs are directed upwards (we then remove the arrow heads).

Example: Consider the poset {1,2, ....,12} with integer division as the partial order. The Hasse diagram for this poset is given by

Def: Let R be a partial order for A and let B be any subset of A. Then aA is an upper bound for B if for every bB, b R a. Also, a is called a least upper bound (or supremum) for B if

  1. a is an upper bound for B, and
  2. a R x for every upper bound x for B.
Similarly, aA is a lower bound for B if for every bB, a R b. Also, a is called a greatest lower bound (or infimum) for B if
  1. a is a lower bound for B, and
  2. x R a for every lower bound x for B.
We write sup(B) [sometimes l.u.b.(B)] to denote the supremum of B and inf(B) [sometimes g.l.b.(B)] to denote the infimum for B.

Examples: In the above example (Hasse diagram), consider the subset B = {2,3}. Both 6 and 12 are upper bounds of this subset and 6 is the sup(B). The only lower bound of B is 1 and inf(B) = 1. Now let B = {4,6}. Sup{B) = 12 and Inf(B) = 2. Consider the set C = {2,3,5}. There is no upper bound for C, and 1 is a lower bound and also inf(C).

Theorem 3.8: If R is a partial order for a set A, and BA, then if sup(B) (or inf(B)) exists, it is unique.

Def: Let R be a partial order for a set A. Let BA. If inf(B) exists and is an element of B, it is called the smallest (or least) element of B. If sup(B) exists and is an element of B, it is called the largest (or greatest) element of B.

Examples: Continuing with our example, we have the subset B = {2,3}. B has no smallest element since inf(B) = 1 and 1 is not in B. On the other hand, the set D = {2,4,6,12} has both a largest element, 12 (since sup(D) = 12) and a smallest element, 2 (since inf(D) = 2). Notice that {2,4,6} would have a smallest element but not a largest element.

Def: A partial ordering R on a set A is called a linear order (or total order) on A if for any two elements x and y of A, either x R y or y R x. (Sometimes we say that every two elements are comparable.)

Examples: Our example is not a total order since, for example, 2 and 11 are not comparable. However, if we restrict the set to {1,2,4,8} then we do have a linear order.

Def: Let L be a linear ordering on a set A. L is a well ordering on A if every nonempty subset B of A contains a smallest element.

Examples: We know that the natural numbers, with the usual ordering, is a well ordered set. Any totally ordered finite set is a well ordered set. The integers, with the usual ordering is not a well ordered set, but if you introduce a different ordering on this set, for instance, use the partial order given by {0,-1,1,-2,2,-3,3,....} then we do get a well ordered set.

Functions

- Definition as a special relation

One-One and Onto (Book uses Range for Image)

Restriction

Composition

Theorem: f is one-one iff f-1 is a function.

if f o g = id then f is onto

if g o f = id then f is one-one

if g and f are onto then f o g is onto.

if g and f are one-one then f o g is one-one, g-1 is one-one. if g is a bijection then g-1 is a bijection.

Permutations

Bijections of a set into itself.

Cyclic notation and digraphs.