Institute of Information Science - University of Miskolc

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==== Relation ====

A **relation** can be narrowly defined as a tool for establishing a connection between two sets. Let \( A \) and \( B \) be two sets. The relation \( R \) creates a connection between these two sets. Let:

\[
C = A \times B = \{(a, b) \mid a \in A, b \in B\}
\]

Here, \( C \) is the **Cartesian product** of sets \( A \) and \( B \). The elements of set \( C \) are ordered pairs from \( A \) and \( B \). The relation \( R \) itself is a set, which is a **subset** of \( C \).

For example, the relation describing the connection between sets \( A \) and \( B \), or the connection between \( A, B, C \to 1, 2, 3, 4 \), can be represented using a **graph** or a **matrix**.

==== Example ====
The relation between the sets \( A, B, C \) and \( 1, 2, 3, 4 \) is shown in the middle diagram. This relationship can also be written using a **matrix**, as shown in the third diagram. Note that there are 1's where the row and column sets are related. The set without a connection will have only 0's in its row.

{{:tanszek:oktatas:techcomm:pasted:20241007-170849.png}}

**Important!**

A relation is **homogeneous** if it is defined on a single set. Relations are generally **binary**, which means that the relation set contains pairs of two elements. Relations are often expressed in the form of statements. For example, "Peter and Irma are married."

A relation can also have an **inverse**. The following diagram shows how a direct relationship can be transformed into its inverse:

{{:tanszek:oktatas:techcomm:pasted:20241007-170952.png}}