16.2.3. Number of Binary Relations

Now an interesting question is that if you have a binary relation defined from the set A to B how many such binary relations can you define? Can I define any number of binary relations or is there an upper bound on the maximum number of binary relations that I can define? Namely how many tables I can form with two columns, where the first column can take entries from the set A and a second column can take entries from the set B.

In this chunk, we are exploring how many binary relations can exist between two sets, A and B. A binary relation is essentially a way to pair elements from two sets. The crucial idea here is to realize that for any given pair of sets A and B, the number of possible binary relations can be calculated. This calculation is based on how subsets of pairs can be formed from the Cartesian product of these two sets.

Imagine you have a box of different colored marbles (representing set A) and a box of different shapes (representing set B). Each marble can be paired with each shape to create a unique combination (pair). The total number of combinations (or binary relations) you can create depends on how many marbles and how many shapes you have. If you have 3 marbles and 4 shapes, you can create 2^(3*4) different sets of pairs! This helps to visualize how many unique relations are possible.

Well it turns out that I can form 2^mn number of binary relations provided A has m number of elements and B has n number of elements. And this simply comes from the observation that you take any relation R, it is nothing but subset of A x B.

This statement explains the mathematical formula for determining the total number of binary relations between two sets A and B. If set A contains 'm' elements and set B contains 'n' elements, the Cartesian product A x B will have m*n unique pairs. Since a relation is a subset of these pairs, every element in the relation can either be included or not. This leads to 2^(mn) possible combinations since each pair can independently either be part of the relation or not. This is an application of the power set concept from set theory.

Let's think about a classroom scenario. Suppose there are 3 students (representing set A) and 2 types of activities (representing set B). Each student can either participate or not in each activity, leading to a total of 2^(3*2) = 64 different ways to form subsets of students for these activities. This example shows how combinations grow rapidly even with a small number of elements in each set.

Now the next question is how do we represent binary relations? So, there are some well-known methods for representing binary relations, the first method is the matrix representation.

Binary relations can be represented using a matrix, which is a grid-like structure containing rows and columns. For a relation R between two sets A and B, with m elements in A and n elements in B, we can create an m x n matrix. The entry in the i-th row and j-th column of this matrix will be '1' if the pair (a_i, b_j) is part of the relation and '0' otherwise. This representation allows for efficient storage and manipulation of relations, especially when dealing with large datasets.

Think of a seating chart in a movie theater. If we have 5 rows (representing elements of set A) and 4 columns (elements of set B) for different types of seats. Every time a seat is filled, you mark it with a '1', and if it’s empty with a '0'. This way, you easily visualize who is sitting where, akin to how a matrix represents a binary relation.

We have another representation which we call as the directed graph representation. So, what do we do in this representation, we draw a graph and by graph I mean a collection of vertices and edges.

Another common way to represent binary relations is through directed graphs. In this representation, each element from the sets A and B becomes a vertex (or node) in the graph. If a relation exists between an element 'a' from set A and an element 'b' from set B, a directed edge is drawn from vertex 'a' to vertex 'b'. This visually displays the relationship and is particularly useful for understanding complex interrelations, such as those found in social networks or computer science algorithms.

Imagine a social network where each person (element from set A) can 'follow' other people (elements from set B). In this graph, if Person A follows Person B, you would draw an arrow from A to B. This visual helps see who influences whom, much like how directed graphs represent relationships in data.

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