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+4 is prime}– 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 ∈∈

• Of special interest is the case A=B.• A relation on a set A is ANY subset R ⊆

A×A

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 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.

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.