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A short class with Sarah and her students. The other ways are under More.
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Today we're going to explore how we can treat relations as sets. Can anyone tell me what set operations we can use?
We can use union, intersection, and difference!
That's correct! Let's start with the union. If we have R1 as 'x < y' and R2 as 'x > y', what do you think R1 union R2 would look like?
It would have pairs where x is not equal to y.
Exactly! So we can say it includes all pairs where x is different from y. Now, what about the intersection of R1 and R2?
That would be an empty set since x cannot be both less than and greater than y at the same time!
Perfect! The intersection is indeed empty here.
Next, let's talk about composing relations. Who can give me the definition of how we define composition?
If R is a relation from A to B and S is from B to C, the composition S o R gives a relation from A to C.
Right! And remember that the order of applying the relations matters. What happens if we switch them?
The composition will likely yield a different result!
Exactly! Composition isn't commutative. Now, can anyone explain what the n+1 power of a relation entails?
It's defined recursively; R to the power of n+1 is the composition of R to the power n with R.
Correct! And we use this to explore transitive relationships as well.
Now, let's connect this to graphs. Can someone explain how an element is related to itself in the m-th power of a relation?
There’s a directed path of length m from the node to itself.
Yes! This means we can visualize relations through paths in a directed graph. How does this help us?
It helps in understanding the transitive nature of relations and connects to powers!
Absolutely! Understanding paths in graphs reinforces our comprehension of relational powers.
Finally, let's discuss the closure of relations. Can anyone summarize what a closure does?
It's the smallest superset of R that satisfies a particular property P.
Spot on! We can consider reflexive, symmetric, and transitive properties. How would we create a reflexive closure?
By taking the union of R with all pairs of the form (a, a) for each a in our set A.
Right you are! And what about symmetric closure?
We take the union of R with its inverse, ensuring that if (a, b) is in R, then (b, a) is too.
Excellent explanation! Closure helps us understand how relations can be expanded while minimizing unnecessary additions.
Combining relations to include pairs from both.
Finding common elements in relations.
Recursive definition through composition.
Paths representing relational connections.
Minimal expansion of relations to satisfy certain properties.
To find the union of sets, make pairs, don't forget. But intersection's the clash, no pairs, what a crash!
Imagine a town where each person represents a relation. They connect through friendship (union) or fight (intersection), but they need a path (composition) to get from one side to the other.
Remember the acronym 'U,I,C' for Union, Intersection, and Composition of relations.
Use 'RAPID' - Relations Are Powers In Dynamics for recalling power concepts in relations.
Term
What is a power of a relation?
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Term
Define closure in relation terms.
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A set of ordered pairs where each pair relates elements from two sets.
Defined recursively as the composition of the relation with itself.
An operation that combines two sets to include all elements from both.
An operation that yields a set of elements common to both sets.
An operation where two relations are combined to illustrate their combined effect.
A property that relates elements through a sequence of relations.
The smallest superset of a relation that satisfies a certain property.
The smallest expansion of a relation to ensure it is reflexive.
The smallest modification of a relation so that if (a, b) is in R, (b, a) is also in R.
A graph where edges are directed, illustrating relationships as arrows.
This section discusses the powers of a relation and operations that can be performed on relations, including union, intersection, and composition.
The section explores how relations can be considered as sets to perform various operations, specifically focusing on the powers of relations, which are defined recursively through compositions. It highlights key concepts such as the transitive paths in directed graphs and provides examples for clarity.
In this section, we delve into the powers of a relation, which are central to understanding concepts in discrete mathematics. A relation can be thought of as a subset of Cartesian products of sets, and by treating relations as sets, we can carry out familiar set operations such as union, intersection, and difference.
Additionally, the composition of two relations R and S is introduced. If R represents a relation from set A to set B and S from B to C, the composition combines these pathways into a relation from A to C, denoted as S o R. The ordering of relations in composition is crucial and not commutative.
The 'n+1'th power of a relation R is defined recursively via composition. For example, R² is the composition of R with itself, encapsulating the transitive relationships established through the paths.
Each relation can be represented as a directed graph. The m-th power of a relation posits that an element is related to itself if there exists a directed path of corresponding length between nodes in the graph.
The section concludes by introducing the closure of a relation concerning specific properties, such as reflexivity, symmetry, and transitivity, emphasizing operations that yield the smallest superset satisfying the desired property.