Binary Relation
2 hr 9 min 20 Examples
- What is a binary relation? Write the relation in roster form (Examples #1-2)
- Write R in roster form and determine domain and range (Example #3)
- How do you Combine Relations? (Example #4a-e)
- Exploring Composite Relations (Examples #5-7)
- Calculating powers of a relation R (Example #8)
- Overview of how to construct an Incidence Matrix
- Find the incidence matrix (Examples #9-12)
- Discover the relation given a matrix and combine incidence matrices (Examples #13-14)
- Creating Directed Graphs (Examples #16-18)
- In-Out Theorem for Directed Graphs (Example #19)
- Identify the relation and construct an incidence matrix and digraph (Examples #19-20)
Discrete Math Relations
1 hr 51 min 15 Examples
- Relation Properties: reflexive, irreflexive, symmetric, antisymmetric, and transitive
- Decide which of the five properties is illustrated for relations in roster form (Examples #1-5)
- Which of the five properties is specified for: x and y are born on the same day (Example #6a)
- Uncover the five properties explains the following: x and y have common grandparents (Example #6b)
- Discover the defined properties for: x divides y if (x,y) are natural numbers (Example #7)
- Identify which properties represents: x + y even if (x,y) are natural numbers (Example #8)
- Find which properties are used in: x + y = 0 if (x,y) are real numbers (Example #9)
- Determine which properties describe the following: congruence modulo 7 if (x,y) are real numbers (Example #10)
- Decide which of the five properties is illustrated given a directed graph (Examples #11-12)
- Define the relation A on power set S, determine which of the five properties are satisfied and draw digraph and incidence matrix (Example #13a-c)
- What is asymmetry? Decide if the relation is symmetric—asymmetric—antisymmetric (Examples #14-15)
Equivalence Relation
1 hr 31 min 17 Examples
- Determine if the relation is an equivalence relation (Examples #1-6)
- Understanding Equivalence Classes – Partitions — Fundamental Theorem of Equivalence Relations
- Turn the partition into an equivalence relation (Examples #7-8)
- Uncover the quotient set A/R (Example #9)
- Find the equivalence class, partition, or equivalence relation (Examples #10-12)
- Prove equivalence relation and find its equivalence classes (Example #13-14)
- Show ~ equivalence relation and find equivalence classes (Examples #15-16)
- Verify ~ equivalence relation, true/false, and equivalence classes (Example #17a-c)
Partial Order
1 hr 10 min 12 Examples
- What is a partial ordering and verify the relation is a poset (Examples #1-3)
- Overview of comparable, incomparable, total ordering, and well ordering
- How to create a Hasse Diagram for a partial order
- Construct a Hasse diagram for each poset (Examples #4-8)
- Finding maximal and minimal elements of a poset (Examples #9-12)
Lattices
1 hr 44 min 9 Examples
- Identify the maximal and minimal elements of a poset (Example #1a-b)
- Classify the upper bound, lower bound, LUB, and GLB (Example #2a-b)
- Find the upper and lower bounds, LUB and GLB if possible (Example #3a-c)
- Draw a Hasse diagram and identify all extremal elements (Example #4)
- Definition of a Lattice — join and meet (Examples #5-6)
- Show the partial order for divisibility is a lattice using three methods (Example #7)
- Determine if the poset is a lattice using Hasse diagrams (Example #8a-e)
- Special Lattices: complete, bounded, complemented, distributed, Boolean, isomorphic
- Lattice Properties: idempotent, commutative, associative, absorption, distributive
- Demonstrate the following properties hold for all elements x and y in lattice L (Example #9)
Chapter Test
1 hr 12 min 6 Practice Problems
- Perform the indicated operation on the relations (Problem #1)
- Determine if an equivalence relation (Problem #2)
- Is the partially ordered set a total ordering (Problem #3)
- Which of the five properties are satisfied (Problem #4a)
- Which of the five properties are satisfied given incidence matrix (Problem #4b)
- Which of the five properties are satisfied given digraph (Problem #4c)
- Consider the poset and draw a Hasse Diagram (Problem #5a)
- Find maximal and minimal elements (Problem #5b)
- Find all upper and lower bounds (Problem #5c-d)
- Find lub and glb for the poset (Problem #5e-f)
- Determine the complement of each element of the partial order (Problem #5g)
- Is the lattice a Boolean algebra? (Problem #5h)
- Is the lattice isomorphic to P(A)? (Problem #5i)
- Show R is an equivalence relation (Problem #6a)
- Find the partition T/R that corresponds to the equivalence relation (Problem #6b)
