Introductory Material

1
Introductory Material

• Review of Discrete Structures up to Lattices

2
Overview

• Sets
• Operations on Sets
• Cartesian Products and Relations
• Order relations
• Lower and upper bounds
• Lattices.

3
Sets

• Will not define set.
• However, everybody (I hope) knows what a set is.
• Described by listing the elements or a common
  property.
• Examples
• Set of people in a room.
• 1,3,4,5,7
• Set of animals in a zoo.
• xx is integer and 3x4 is prime
• etc

4
Relations between sets

• Let A and B be two sets.
• If every element of A is an element of B we say
  that A is a subset of B and write A?B
• If A is a subset of B and B is a subset of A,
  then AB
• There is a special set Ø which does not contain
  any elements. It is a subset of every set.

5
Operations on Sets

• Let A and B be two sets. The union or join of A
  and B, A?B is the collection which contains all
  the elements from both A and B.
• Let A and B be two sets. The intersection or meet
  of A and B, AnB is those elements which are in
  both A and B. It is perfectly OK for there not to
  be any such sets are called disjoint.
• Let A and B be two sets. The set difference, A-B
  is the collection of those elements of A which
  are not in B.

6
Cartesian Product and Relations

• Let A, B be two sets. The Cartesian Product of A
  and B is a collection of all pairs where the
  first element in the couple belongs to A and the
  second to B.
• AB (a,b), a ? A, b ? B
• Of special interest is the case AB.
• A relation on a set A is ANY subset R ? AA

7
Properties in Relations

• There are some relations that are more
  interesting than others, because they satisfy
  certain properties. For example
• Reflexive For all x in A, xRx.
• Transitive For all x,y, z in A, if xRy and yRz,
  then xRz.

8
Order Relations

• A (partial) order relation is a relation which is
  reflexive, transitive, and antisymmetric
• For any x,y in A if xRy and yRx then xy.
• Examples order between numbers, containment
  between sets, divisibility between positive
  numbers, etc.
• A set with a partial order is called a partially
  ordered set.
• A partial order which satisfies, for any a,b
  either aRb or bRa, is called total.

9
Lower and Upper Bounds

• Let A be a set with a partial order R. Given two
  elements a,b of A, a lower bound l of a and b is
  an element satisfying lRa and lRb.
• If among all the lower bounds of a and b there is
  one that is bigger than all the others, that
  element is called the greatest lower bound of a
  and b.
• We can similarly define least upper bound.
• Sometimes, the glb is called the meet and the
  lub is called the join of the two elements.

10
Lattices

• A lattice is a partially ordered set in which any
  two elements have a glb and lub.
• A lattice is complete if every subset has a glb
  and an lub.
• Note that any finite lattice is complete.