Introductory Material Review of Discrete Structures up to Lattices

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

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. {x:x is integer and 3x+5 is even} etc

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 A=B There is a special set Ø which does not contain any elements. It is a subset of every set.

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, A∩B 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.

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. A×B = {(a,b), a ∈ A, b ∈ B} Of special interest is the case A=B. A relation on a set A is ANY subset R ⊆ A×B

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.

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 x=y. 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.

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 greater 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.

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.