יחסים ופונקציות - 3 - חזקות של יחס

Summary

This video explores the concept of composition and exponentiation of relations, particularly focusing on relations from a set to itself. It explains how relations can be composed, how powers of a relation are defined inductively, and how the identity relation behaves within this context. The video also delves into negative exponents and the inverse of relations, emphasizing the differences between positive and negative exponentiation. Key concepts include the behavior of inverse relations, the importance of composition order, and the challenges that arise with negative powers of relations.

Takeaways

• 😀 Relation composition involves combining two relations to form a new one, where pairs like (a, c) exist if there’s an intermediate b such that (a, b) and (b, c) are in the composed relations.
• 😀 A relation R from set A to itself is represented as a relation above set A, where arrows connect elements of A to themselves.
• 😀 Composition of relations requires one relation from A to B and another from B to C, where they share set B. When dealing with relations over A, any pair of relations can be composed.
• 😀 The concept of exponentiation of relations allows for compositions like R^2 (R composed with itself), R^3, R^4, etc.
• 😀 Exponentiation of relations is defined by induction: R^0 is the identity relation, and R^(n+1) is the composition of R^n and R.
• 😀 The exponentiation of relations is designed to behave similarly to regular exponentiation, where R^n composed with R^m equals R^(n+m).
• 😀 The identity relation (IA) is a set of pairs (a, a) for each element a in A. It acts as the neutral element in relation composition, similar to 1 in multiplication.
• 😀 R^0 is defined as the identity relation, IA, and any relation composed with IA remains unchanged, like multiplying by 1.
• 😀 Negative exponents of relations are defined differently. R^-1 is the inverse of relation R, and its further exponents follow the exponents of R^-1.
• 😀 The composition of a relation with its inverse does not always yield the identity relation, as seen in examples where the composition misses some required pairs.
• 😀 The inverse of a relation R can be defined as a general relation from A to B, even when A and B are different sets. The inverse of the composition of relations is the composition of their inverses in reverse order.

Q & A

What is the concept of composition of two relations as described in the script?

The composition of two relations means creating a new relation such that if pair (a, c) is in this relation, there must exist a pair (a, b) in one relation and (b, c) in another relation, where b is a common element between the two relations.

What is the specific case discussed regarding a relation R from a set A to itself?

In the case where the relation R is from set A to itself, the relation is said to be 'above' set A. The diagram used shows set A once, and the arrows are drawn between the elements of A itself.

What is meant by the composition of relation R with itself, and how is it denoted?

The composition of relation R with itself is called 'R squared'. It refers to the composition of R with R, which creates a new relation over the same set A.

What is the objective of defining powers of relations, and how is it linked to normal exponentiation?

The goal is to define the exponentiation of relations in such a way that it behaves similarly to normal powers in regular multiplication, specifically that the composition of R raised to the power of n and R raised to the power of m equals R raised to the power of (n + m).

What is the relation R raised to the power of 0, and how does it behave in the context of composition?

R to the power of 0 is defined as the identity relation (IA), which contains pairs (a, a) for all elements a in set A. This relation behaves like multiplying by 1 in normal arithmetic; when composed with any other relation, it leaves the other relation unchanged.

How is R raised to the power of n+1 defined, and what does this imply for the composition of relations?

R to the power of n+1 is defined as the composition of R raised to the power of n with R. This recursive definition ensures that exponentiation of relations can be extended to any natural number n.

What is the definition of negative powers of relations, and how does it differ from positive exponents?

Negative powers of relations are defined by first taking the inverse of the relation R. For example, R to the power of -1 is the inverse of R. The negative exponentiation follows the same structure as positive exponents but is applied to the inverse relation.

Why must we be cautious when applying the composition law to negative exponents of relations?

The composition law for exponents (i.e., R^n composed with R^m equals R^(n+m)) does not necessarily hold for negative exponents. This is because the composition of a relation with its inverse may not yield the identity relation, as shown in the example where R is the pair (1, 2) and its inverse is (2, 1).

How is the inverse of a relation R defined and what are its properties?

The inverse of a relation R is formed by reversing the direction of its pairs. For instance, if R contains the pair (a, b), the inverse relation contains (b, a). This inverse relation can be composed and exponentiated like any other relation.

What is the inverse of the composition of two relations, and how is it related to the order of the relations?

The inverse of the composition of two relations, R and S, is the composition of the inverses of R and S, but with the order reversed. Specifically, the inverse of R composed with S is S to the power of -1 composed with R to the power of -1, with S's inverse coming first.