Relations, properties of relations. Equivalence relations. Partial orderings презентация

Relations

Relations

Relations

Relations

Relations
We can use the concept of a directed graph to describe the ordered pairs belonging to a given binary relation.

Relations

1 2
3 4
5 6
7

Relations

Properties of relations

Properties of relations

Properties of relations

Properties of relations

Properties of relations

Properties of relations

Properties of relations

Properties of relations

Properties of relations

Properties of relations

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Equivalence relation

Partial Orderings

Partial Orderings

Partial Orderings
Example 2
The divisibility relation| is a partial ordering on the set of positive integers, because it is reflexive, antisymmetric, and transitive.

Partial Orderings

Partial Orderings

Partial Orderings

Partial Orderings

Partial Orderings

Partial Orderings

Start with the directed graph for this relation.
3
4
2
1

Remove all loops.
3
4
2
1

Remove all loops.
3
4
2
1

3
4
2
1

3
4
2
1

Remove all the arrows on the directed edges, because all edges point “upward” toward their terminal vertex.
3
4
2
1

Remove all the arrows on the directed edges, because all edges point “upward” toward their terminal vertex.
3
4
2
1

12
6
3
1
8
12
4
2

Maximal and minimal elements

Maximal and minimal elements
Maximal and minimal elements are easy to spot using a Hasse diagram.
They are the “top” and “bottom” elements in the diagram.

The maximal elements are 8, 12, and the minimal element is 1.
12
6
3
1
8
12
4
2

Topological sorting

Topological sorting

Topological sorting

The topological sorting algorithm

The topological sorting algorithm

The topological sorting algorithm
20
2
5
20
12
4
1

The topological sorting algorithm
20
2
5
20
12
4
1

The topological sorting algorithm
20
2
5
20
12
4

The topological sorting algorithm
20
2
5
20
12
4

The topological sorting algorithm
20
2
20
12
4

The topological sorting algorithm
20
2
20
12
4

The topological sorting algorithm
20
20
12
4

The topological sorting algorithm
20
20
12
4

The topological sorting algorithm
20
20
12

The topological sorting algorithm
20
20
12

The topological sorting algorithm
12

The topological sorting algorithm

The topological sorting algorithm

The topological sorting algorithm
Example 8
Find a compatible total ordering for the poset.
12

The topological sorting algorithm
12

The topological sorting algorithm
12

The topological sorting algorithm
12

The topological sorting algorithm
12

The topological sorting algorithm
12

The topological sorting algorithm
12

The topological sorting algorithm
12

The topological sorting algorithm
12

The topological sorting algorithm
12

The topological sorting algorithm
12

The topological sorting algorithm
12

The topological sorting algorithm
12