Question

There are 16 possible different relations \(R\) on the set \(A=\\{a, b\\} .\) Describe all of them. (A picture for each one will suffice, but don't forget to label the nodes.) Which ones are reflexive? Symmetric? Transitive?

Short answer

There are 16 possible relations, categorized into reflexive, symmetric, and transitive relations. For example, \(\{(a, a), (b, b)\}\) is reflexive, and \(\{(a, b), (b, a)\}\) is symmetric. Other relations may exhibit one, two, or all three of these properties, or none at all. The categorization depends on whether the relation satisfies the mathematical definitions of reflexivity, symmetry, and transitivity.

Step-by-step solution

Enumerate all possible relations

To start, establish the definition of a binary relation from set \(A\). Then describe the 16 possible relations \(R\), as captured in the power set of \(A \times A\). Each of these can be listed as a combination of elements \(\{(a, a), (a, b), (b,a), (b, b)\}\). For example, the following are some of the relations: \(\{\}, \{(a, a)\}, \{(b, b)\}, \{(a, b)\}, \{(b, a)\}, \{(a, a), (b, b)\}\), and so on.

Apply definitions of reflexive, symmetric, and transitive

A relation \(R\) on \(A\) is reflexive if \(\forall x \in A, (x,x) \in R\). It is symmetric if, \(\forall x, y \in A, (x,y) \in R\) implies that \( (y,x) \in R\). It is transitive if, \(\forall x, y, z \in A, (x,y) \in R\) and \((y, z) \in R\) imply \((x,z) \in R\). Evaluate each of the 16 relations by these definitions.

Categorize each relation

Use the definitions of reflexive, symmetric, and transitive to categorize each of the 16 relations. For example, relation \(\{(a, a), (b, b)\}\) is reflexive, but it is not symmetric or transitive. On the other hand, relation \(\{(a, b), (b, a)\}\) is symmetric but not reflexive or transitive.

Key concepts

Set Theory

Set theory is the mathematical study of collections of objects, which are called sets. Each set is considered as a single entity, capable of containing other entities, and these can be anything from numbers to letters, or even other sets. Sets are denoted using curly braces \( \{ \} \), with elements listed inside, such as \( A = \{a, b\} \).

Understanding set theory is crucial when delving into more complex mathematical concepts, as it forms the foundation for binary relations and various operations such as union, intersection, and complement. In the context of binary relations, one fundamental operation is the Cartesian product, denoted by \(\times\). For example, the Cartesian product of set \(A\), \(\times\) itself, denoted by \(\times A\), is the set of all possible ordered pairs where the first and second elements are from \(A\). In the case of the given exercise, \(A \times A\) yields the set \(\{(a, a), (a, b), (b, a), (b, b)\}\), from which all possible relations on \(A\) can be formed.

Reflexive Relation

A reflexive relation is a type of binary relation where for all elements in the set, each element is related to itself. In formal terms, a relation \(R\) on a set \(A\) is reflexive if for every \(x \) in \(A\), the ordered pair \( (x, x) \) is in \(R\).

For instance, considering the set \(A = \{a, b\}\), a relation \(R\) is reflexive if both \( (a, a) \) and \( (b, b) \) are elements of \(R\). It represents a concept of self-association within the set. In our exercise, the relation \( \{(a, a), (b, b)\} \) is an example of a reflexive relation, as it satisfies the condition that each element is paired with itself.

Symmetric Relation

A symmetric relation is another variety of binary relation characterized by a bidirectional relationship between the elements. If an element \(a\) is related to another element \(b\), then \(b\) must also be related to \(a\). In more formal terms, a relation \(R\) on a set \(A\) is symmetric if whenever \( (x, y) \) is a member of \(R\), the reversed pair \( (y, x) \) is also a member of \(R\), for all \(x, y \) in \(A\).

In the exercise provided, the relation \( \{(a, b), (b, a)\} \) is an illustration of a symmetric relation because not only is \(a\) related to \(b\), but \(b\) is also related to \(a\). Symmetry is an important concept in relations because it ensures that the relationship works both ways.

Transitive Relation

The concept of transitivity in set theory refers to a relation that maintains consistency across a chain of connections between elements. Specifically, a relation \(R\) on a set \(A\) is transitive if whenever an element \(a\) is related to \(b\) and \(b\) is related to \(c\), then \(a\) must also be related to \(c\). Formally, for all elements \(x, y, z \) in \(A\), if \( (x, y) \) and \( (y, z) \) are in \(R\), it follows that \( (x, z) \) must be in \(R\) as well.

Looking at our exercise's set \(A = \{a, b\}\), identifying transitive relations involves checking for chains where two pairs share an element and ensuring the presence of a third pair that links the first and last elements of the chain. For example, if a relation contained the pairs \( (a, b) \) and \( (b, a) \) but not the pair \( (a, a) \) or \( (b, b) \) it would not be transitive, since the chain is incomplete.