*Equivalence Classes*

*Partitions*

*Examples:*

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

**Def**: A relation R on a set A is a ** partial order** (or

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

- a is an upper bound for B, and
- a R x for every upper bound x for B.

- a is a lower bound for B, and
- x R a for every lower bound x 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)

*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

*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.

*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.