18.2. Intersection of Relations

Let us consider two relations R1 and R2, where R1 consists of all (x, y) pairs where x < y, and R2 consists of all (x, y) pairs where x > y. The union of these two relations will have all pairs of the form (x, y) where x is not equal to y.

In the context of set theory, the union of two sets combines all elements from both sets without duplication. In our case, we are looking at two relations, R1 and R2. R1 consists of pairs where the first element is less than the second (x < y), while R2 consists of pairs where the first element is greater than the second (x > y). When we take the union of these relations, we obtain all possible pairs (x, y) such that x is not equal to y. This means we can have pairs where x is less than y from R1 and pairs where x is greater than y from R2 combined into one relation.

Think of relations as two different job conditions for candidates. In R1, we might have candidates applying for jobs where their age is less than a certain limit (e.g., 'y'). In R2, we have candidates who are older than that limit. The union of these two sets would include all candidates who are either younger than or older than that age, effectively considering all candidates except the ones who fall exactly on the age limit.

When taking the intersection of R1 and R2, we find an empty set, as there cannot be any real numbers x and y where x is both less than and greater than y simultaneously.

The intersection of two sets is defined as the elements that are common to both sets. For our relations R1 and R2, which describe all pairs (x, y) with x < y and x > y respectively, there cannot be any pair (x, y) that satisfies both conditions at the same time. This is because if x is less than y, it cannot simultaneously be greater than y. Therefore, the intersection of these two relations yields an empty set, as there are no pairs to be found in both relations.

Imagine two groups of people: one group that is younger than age 18, and another group that is older than age 18. There cannot be anyone who is both younger than 18 and older than 18 at the same time, hence the intersection of these two groups is empty, just as the intersection of our two relations is empty.

The difference of R1 and R2 will return R1 itself, which consists of all (x, y) pairs where x is less than y.

The difference between two sets, often denoted as A - B, includes elements that are in set A but not in set B. In our case, when we take the difference of R1 (all pairs where x < y) and R2 (all pairs where x > y), we end up with R1 itself because there are no elements in R2 that can be subtracted from R1. Since R2 includes pairs that represent a completely opposite condition (x > y), it does not affect the original relation R1 at all.

Think about a basket containing apples and another basket containing oranges. If you’re asked to find the apples that are not oranges, you’ll simply end up with all the apples since oranges have no overlap with apples at all. Here, the apples represent elements in R1, and the oranges represent elements in R2, resulting in your finding all apples untouched.

The composition of relations R and S is denoted as S o R, which means we apply relation R first followed by relation S. If R relates set A to set B, and S relates set B to set C, then the result is a relation from A to C.

Composition of relations involves combining two relations such that you link the sets based on their relationships. For instance, if relation R connects elements from set A to B (denoted as R ⊆ A x B) and relation S connects elements from B to C (denoted as S ⊆ B x C), then the composition S o R results in a new relation connecting A directly to C. The key here is that S o R is not the same as R o S; the order of composition matters because you first apply R and then S to find the new relation from A to C.

Imagine a scenario where you have a set of students (set A) who are taking courses (set B), and those courses are linked to various textbooks (set C). If the first relation (R) describes which student is taking which course, and the second (S) describes which course uses which textbook, then the composition of these two relations will tell you which student is associated with which textbook. The order is vital: you first find out the courses students are taking, and then you find out the textbooks for those courses.

If R is a relation from A to B, R1 is said to be R itself, and the (n+1)th power of R is defined as the composition of the relation Rn with the relation R.

The powers of a relation can be thought of as repeated applications of the relation. For example, R^n represents applying the relation R successively n times. The base relation R^1 is simply R itself, while R^2 is formed by composing R with itself once (i.e., R o R). This process is recursive; the (n+1)th power consists of the composition of Rn with R, emphasizing that the order is important – you apply R first before combining it again with Rn.

Consider a relay race where team members pass a baton. If the first runner represents the first application of the relation R (let's say the runner who starts the race), then the second runner who receives the baton represents the second application. Continuing this process with more runners can be thought of as R squared, R cubed, etc. Each runner in the race is analogous to successive powers of the relation, demonstrating how one can build upon previous connections.

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