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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 exploring how we can treat relations as sets! Can anyone tell me what set operations we know?
Union and intersection!
Exactly! Let's consider two relations R1 and R2 where R1 contains pairs (x, y) such that x < y, and R2 contains (x, y) such that x > y. What do you think happens if we take the union of these two relations?
It would include all pairs where x is not equal to y.
Correct! So, the union has all (x, y) pairs where x is different from y. What about the intersection? What do you think we would get?
An empty set, because x can't be both less than and greater than y at the same time.
Well done! The intersection indeed results in an empty set. Let's summarize: The union brings together different pairs while the intersection shows us when neither holds.
Now, let’s shift gears and talk about composition of relations. If R is a relation from A to B and S from B to C, can anyone explain how we would write the composition?
It would be written as S o R, right? We apply R first and then S!
Exactly! This gives the ordered pairs of the form (a, c). If a is related to b in R and b is related to c in S, then a is related to c in S o R. Why do you think this order is important?
Because if we did R o S, it would mean something different!
Absolutely! Order is critical in relations. So remember, S o R is different from R o S. Let's wrap this up: composition allows us to connect relations across sets!
Next, let’s discuss the powers of relations. If I have a relation R, what do we mean by R squared or R cubed?
R squared would be the composition of R with itself.
Right! And it follows recursively. Can anyone define R to the power of n?
It’s the composition of R with itself n times?
Exactly! Remember though, the order matters. R squared is not the same as squaring R and it might not be equal to R o R. Can someone think of a scenario where this would matter?
If the relations are not commutative! Like multiplication.
Great analogy! Non-commutativity makes a significant difference in the results.
Now let’s visualize relations. How does representing relations on directed graphs help us understand them better?
We can see the connections between elements more clearly!
Exactly! If we have a directed path of length m between nodes, it relates to powers of the relation. How so?
If there's a directed path from a to a of length m, it shows that the ordered pair (a, a) is present in that power.
Right! Visualizing these relations helps in grasping the concept of paths and powers better. In summary, graphical representations offer crucial insights into our relations!
Lastly, let’s talk about closure of relations. What does it mean when we say closure under certain properties like reflexivity?
It's about expanding the relation to satisfy that property, right?
Good point! If relation R isn’t reflexive, we’ll add pairs of the form (a, a) for each a in set A. What about symmetry?
We check if (a, b) exists, then we should add (b, a) if it's not there.
Exactly! To ensure symmetry, we take the union of R and its inverse. So to conclude, closures help us adjust relations to meet specific properties.
Combines all pairs from both relations.
The common pairs from both relations, often resulting in an empty set for contrasting relations.
Connects two relations by chaining them together based on shared elements.
Recursive application of relations to establish transitive connections.
Adjusts relations to meet defined properties such as reflexivity, symmetry, or transitivity.
Pairs unite, get in the fight, union brings them all to light.
Once a pair went looking for friends. Together they formed alliances in a union that never ends.
U-I-D for Union, Intersection, and Difference helps to remember the main operations!
C-R-P for Closure means Reflexive, Symmetric, and Transitive properties to remember.
Term
What is the union of relations?
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Define the composition of relations.
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What does the transitive closure ensure?
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What is the reflexive closure?
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The combination of all ordered pairs from both relations.
The set of ordered pairs common to both relations.
The ordered pairs in the first relation that are not in the second.
A relation derived from two relations where the elements of one relation connect with the elements of another.
Repeated application of a relation; defined recursively.
The smallest extension of a relation to satisfy a given property.
A relation where every element is related to itself.
A relation where if (a, b) exists, then (b, a) also exists.
A relation where if (a, b) and (b, c) exist, then (a, c) must also exist.
This section discusses important operations on relations including set operations (union, intersection, difference) and introduces the concepts of composition of relations and their powers.
In this section, we explore operations that can be performed on relations, such as union, intersection, difference, and the composition of relations. We also define the powers of a relation and understand these concepts through examples, demonstrating their significance in relational mathematics.
In the study of discrete mathematics, particularly in relation to set theory, understanding relations is crucial. This section delves into the operations one can perform on relations, some of which include:
Relations can be treated as sets, enabling the use of set theoretic operations:
The section introduces the concept of composing relations, denoted by S o R, where R is a relation from set A to set B, and S from B to C. Composition is performed by connecting related elements from each set:
Powers of a relation are defined recursively. R1 is R itself, and R(n+1) is the composition of R(n) with R. The order of composition matters, as it does not imply commutativity.
Directed graphs represent relations, where nodes signify elements and directed edges indicate the relationships. The presence of a directed path of length m between two nodes corresponds directly to the powers of relations.
Further explained is the concept of closure involving certain properties like reflexivity, symmetry, and transitivity, with examples demonstrating the construction of various types of closures.
Overall, this section is pivotal for understanding more complex concepts in mathematics related to relations and their operations. The examples provided lay a foundational understanding that will be built upon in subsequent sections.