BCA PART A 6.R program to check whether the given relation is transitive.

 

A relation is said to be transitive if, whenever a first element is related to a second element, and the second element is related to a third element, then the first element must also be related to the third element.

Formal Definition:

A relation $R$ on a set $A$ is transitive if, for all elements $a$, $b$, and $c$ in $A$, whenever $(a, b) \in R$ and $(b, c) \in R$, then $(a, c)$ must also be in $R$.

Imagine you have three friends: Alice, Bob, and Charlie. If Alice is friends with Bob, and Bob is friends with Charlie, a transitive relationship would mean that Alice must also be friends with Charlie. If this condition is always true, the friendship relation is transitive.

Examples:

1. Transitive Relation Example:

   • Consider the relation "is greater than" on numbers. If $a > b$ and $b > c$, then $a > c$. This is a transitive relation.

2. Non-Transitive Relation Example:

• Consider the relation "is the parent of" on people. If Alice is the parent of Bob, and Bob is the parent of Charlie, Alice is not necessarily the parent of Charlie. Therefore, "is the parent of" is not a transitive relation.

Program

# Function to check transitivity of a relation

is_transitive <- function(relation) {

  # Flatten the relation list to get all elements and find the maximum element

  elements <- unlist(relation)

  n <- max(elements)

  

  # Initialize an n x n matrix with zeros

  relation_matrix <- matrix(0, n, n)

  

  # Populate the matrix with 1s based on the relation pairs

  for (pair in relation) {

    relation_matrix[pair[1], pair[2]] <- 1

  }

  

  # Check transitivity

  for (a in 1:n) {

    for (b in 1:n) {

      if (relation_matrix[a, b] == 1) {

        for (c in 1:n) {

          if (relation_matrix[b, c] == 1 && relation_matrix[a, c] == 0) {

            return(FALSE)

          }

        }

      }

    }

  }

  

  return(TRUE)

}

# Example relation as a list of pairs

relation <- list(c(1, 2), c(2, 3), c(1, 3))

# Check if the relation is transitive

if (is_transitive(relation)) {

  print("The relation is transitive.")

} else {

  print("The relation is not transitive.")

Output

Rscript /tmp/qu6SQFlEND.r

[1] "The relation is transitive."

Explanation

1. Function Definition (is_transitive)

• The program defines a function called is_transitive that takes a relation as input. This relation is a list of ordered pairs, where each pair represents a connection between two elements.

2. Matrix Representation

• The program first determines the maximum element in the relation to create a square matrix (of size n x n). This matrix (relation_matrix) is initialized with zeros.
• The matrix represents the relation: if there's a pair $(a, b)$ in the relation, the matrix entry at row a and column b is set to 1.

3. Transitivity Check

• The program then checks for transitivity. For each pair $(a, b)$ in the matrix, it looks for pairs $(b, c)$.
• If both $(a, b)$ and $(b, c)$ exist (i.e., are 1 in the matrix), the program checks if $(a, c)$ is also present (i.e., if relation_matrix[a, c] is 1).
• If any such $(a, c)$ pair is missing, the function returns FALSE, indicating that the relation is not transitive.

4. Output

• If the relation passes the transitivity check for all possible combinations, the function returns TRUE.
• The main program checks the result of the function and prints whether the relation is transitive.

Example Explanation:

• For relation <- list(c(1, 2), c(2, 3), c(1, 3)), the matrix would have entries at (1,2), (2,3), and (1,3) set to 1.
• The program confirms that since $(1, 2)$ and $(2, 3)$ exist, $(1, 3)$ must also exist, satisfying transitivity. Hence, it outputs "The relation is transitive."

This program helps to verify if a given set of ordered pairs follows the transitivity property in mathematics.