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Fundamentals of Discrete Structures: Sets, Relations, and Lattices Overview

Introduction to sets, operations on sets, Cartesian products, relations, order relations, lower/upper bounds, and lattices in discrete structures. Descriptions and examples provided in detail.

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Fundamentals of Discrete Structures: Sets, Relations, and Lattices Overview

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  1. Review of Discrete Structures up to Lattices Introductory Material

  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. • {x:x is integer and 3x+4 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 A=B • 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, 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.

  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. • 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×A

  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 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.

  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.

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