Matrix Representation of a Relation in Discrete Mathematics
In Discrete Mathematics, relations can be represented in different ways such as ordered pairs, digraphs, and matrices. Among these, the matrix representation of a relation is very useful for performing operations like union, intersection, complement, and composition of relations.
Matrix Representation of a Relation
Let
$A = \{a_1, a_2, a_3, \dots, a_n\}$
$B = \{b_1, b_2, b_3, \dots, b_m\}$
be two finite sets containing $n$ and $m$ elements respectively. Then the Cartesian product $A \times B$ contains $n \times m$ ordered pairs.
Let $R$ be a relation from set $A$ to set $B$. Then,
$$ R \subseteq A \times B $$
The matrix of relation $R$, denoted by $M_R$, is an $n \times m$ matrix defined as follows:
$$ M_R = [m_{ij}] $$
where
- $m_{ij} = 1$ if $(a_i, b_j) \in R$
- $m_{ij} = 0$ if $(a_i, b_j) \notin R$
Question 1
Let
$A = \{1,2,3,4\}$
$B = \{1,4,6,8,9,16\}$
A relation $R$ from $A$ to $B$ is defined by $aRb$ if and only if $b = a^2$.
Find the matrix representation of the relation $R$.
Solution
Step 1: Write the relation explicitly
We check each element of $A$ and find its square in $B$.
- $1^2 = 1 \in B \Rightarrow (1,1) \in R$
- $2^2 = 4 \in B \Rightarrow (2,4) \in R$
- $3^2 = 9 \in B \Rightarrow (3,9) \in R$
- $4^2 = 16 \in B \Rightarrow (4,16) \in R$
Therefore, the relation is
$$ R = \{(1,1),(2,4),(3,9),(4,16)\} $$
Step 2: Arrange rows and columns
Rows correspond to elements of set $A$ in the order:
$$ 1,\;2,\;3,\;4 $$
Columns correspond to elements of set $B$ in the order:
$$ 1,\;4,\;6,\;8,\;9,\;16 $$
Step 3: Construct the relation matrix
We place 1 if the ordered pair exists in $R$, otherwise 0.
$$ M_R = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} $$
This is a $4 \times 6$ relation matrix.
Final Answer
The matrix representation of the given relation $R$ is
$$ M_R = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} $$
Important Notes
- Rows always represent elements of the first set.
- Columns always represent elements of the second set.
- Matrix representation is useful for relation operations.
Keywords
Question 2
Let
$A = \{1,2,3,4,6\}$ and $B = \{1,2,3,4,6\}$.
A relation $R$ on $A$ is defined by $aRb$ if and only if $a$ divides $b$.
Find the matrix representation of the relation $R$.
Solution
Step 1: Write the relation explicitly
We list all ordered pairs $(a,b)$ such that $a$ divides $b$.
$$ R = \{(1,1),(1,2),(1,3),(1,4),(1,6), (2,2),(2,4),(2,6), (3,3),(3,6), (4,4), (6,6)\} $$
Step 2: Fix the order of elements
Rows and columns are taken in the order:
$$ 1,\;2,\;3,\;4,\;6 $$
Step 3: Construct the relation matrix
Place 1 if $(a,b) \in R$, otherwise 0.
$$ M_R = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} $$
This is a $5 \times 5$ matrix.
Final Answer
The matrix representation of the relation $R$ is given above.
Question 3
Let
$A = \{1,2,3,5,6,10,15,30\}$ and $B = \{1,2,3,5,6,10,15,30\}$.
A relation $R$ on $A$ is defined by $aRb$ if and only if $a$ divides $b$.
Find the matrix representation of the relation $R$.
Solution
Step 1: Understanding the relation
For each element $a \in A$, we check which elements of $B$ are divisible by $a$.
Step 2: Fix the order of elements
Rows and columns are arranged in the order:
$$ 1,\;2,\;3,\;5,\;6,\;10,\;15,\;30 $$
Step 3: Construct the relation matrix
Using the definition of divisibility, the relation matrix is:
$$ M_R = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} $$
This is an $8 \times 8$ relation matrix.
Final Answer
The matrix $M_R$ represents the given relation defined by divisibility.
Important Observations
- The diagonal entries are always 1 because every number divides itself.
- The matrix is upper triangular due to divisibility ordering.
- Relation matrices are useful for computing transitive closures.
Applications of Matrix Representation of a Relation
The matrix representation of a relation is not only a mathematical tool, but also a very powerful concept from the point of view of computers and real-world applications. It connects abstract mathematics with computer science and digital technology.
1. Why Matrix Representation is Important
In mathematics, a relation is defined as a set of ordered pairs. However, computers do not understand sets or ordered pairs directly. A computer processes information in the form of numbers, especially binary digits.
A relation matrix contains only two values:
- 1 – the relation exists
- 0 – the relation does not exist
This makes the matrix representation perfectly suitable for computers, because computers internally work using binary language (0 and 1).
2. Relation Matrix and Binary Language
Every entry in a relation matrix answers a simple yes-or-no question:
“Does element a relate to element b?”
If the answer is Yes, the matrix entry is 1. If the answer is No, the matrix entry is 0.
This binary decision-making process is exactly how computers work. Therefore, relation matrices allow relations to be stored, processed, and analyzed efficiently by computers.
3. Use in Computer Algorithms
Many algorithms in computer science are based on matrix representations. Once a relation is written in matrix form, it becomes easy to:
- Find union and intersection of relations
- Compute composition of relations
- Determine transitive closure using algorithms
- Perform fast computations using matrix operations
These operations are much faster on matrices than on sets of ordered pairs, especially when the data size is large.
4. Application in Graph Theory
In graph theory, relations are often represented using directed graphs. The same information can also be represented using a relation matrix.
In fact, the relation matrix is closely related to the adjacency matrix of a graph.
This is widely used in:
- Network analysis
- Communication systems
- Social network modeling
- Routing and connectivity problems
5. Use in Databases and Information Systems
Databases store relationships between data items such as:
- Student – Course relationships
- User – Permission relationships
- Product – Category relationships
Internally, such relationships are often converted into binary tables or matrices, where 1 indicates the presence of a relationship and 0 indicates its absence.
Thus, matrix representation helps in efficient data storage and quick data retrieval.
6. Real-Life Interpretation
Consider a simple example:
“Is person A connected to person B on a social media platform?”
This question can be answered using a binary value:
- 1 – Connected
- 0 – Not connected
Such real-life relations are naturally modeled using matrices, making the matrix representation a bridge between mathematics and real-world systems.
7. Summary
The matrix representation of a relation is powerful because:
- It uses only 0 and 1, which computers understand naturally
- It allows fast and efficient computation
- It connects relations with graphs and algorithms
- It has wide applications in computer science and data systems
Hence, the matrix representation of a relation plays a vital role in both theoretical mathematics and practical computing applications.
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