BCA PART A 8.R Program to Check if a Relation is Symmetric:

 

A symmetric relation is a type of mathematical relation on a set where the order of elements in the pairs does not matter. Specifically, a relation $R$ on a set $A$ is called symmetric if, for every pair $(a, b) \in R$, the reverse pair $(b, a)$ is also in $R$.

Formal Definition:

A relation $R$ on a set $A$ is symmetric if: $\forall a, b \in A, \, (a, b) \in R \Rightarrow (b, a) \in R$ This means that if an element $a$ is related to an element $b$, then $b$ must also be related to $a$.

Example:

• Set $A$: $\{1, 2, 3\}$
• Relation $R$: $\{(1, 2), (2, 1), (3, 3)\}$

In this case, the relation $R$ is symmetric because:

• $(1, 2)$ is in $R$, and the reverse pair $(2, 1)$ is also in $R$.
• $(3, 3)$ is its own reverse, so it satisfies the condition.

Non-Symmetric Example:

• Set $A$: $\{1, 2, 3\}$
• Relation $R$: $\{(1, 2), (3, 3)\}$

In this case, the relation $R$ is not symmetric because:

• $(1, 2)$ is in $R$, but the reverse pair $(2, 1)$ is not in $R$, so the relation is not symmetric.

# Function to check if a relation is symmetric

is_symmetric <- function(relation) {

  # Iterate through each pair in the relation

  for (i in 1:nrow(relation)) {

    # Get the current pair (a, b)

    a <- relation[i, 1]

    b <- relation[i, 2]

   

    # Check if the pair (b, a) is in the relation

    if (!any(relation[,1] == b & relation[,2] == a)) {

      return(FALSE)

    }

  }

  return(TRUE)

}

# Example usage

# Define the relation R as a matrix of pairs

relation <- matrix(c(1, 2,

                     2, 1,

                     3, 3),

                   ncol = 2, byrow = TRUE)

# Check if the relation is symmetric

if (is_symmetric(relation)) {

  print("The relation is symmetric.")

} else {

  print("The relation is not symmetric.")

}

The program checks whether a given relation is symmetric. A relation is symmetric if for every pair $(a, b)$, the pair $(b, a)$ also exists.

Explanation :

1. Function is_symmetric:

   • The function takes one argument, relation, which is a matrix of pairs representing the relation.
   • It iterates through each pair $(a, b)$ in the matrix using a loop.
   • For each pair $(a, b)$, the function checks if the reverse pair $(b, a)$ is also present in the relation.
   • If any reverse pair $(b, a)$ is missing, the function returns FALSE (indicating the relation is not symmetric).
   • If all pairs have their reverse counterparts, it returns TRUE (indicating the relation is symmetric).

2. Example Relation:

   • The relation is represented as a matrix, e.g., relation <- matrix(c(1, 2, 2, 1, 3, 3), ncol = 2, byrow = TRUE).
   • This relation contains pairs $(1, 2)$, $(2, 1)$, and $(3, 3)$.

3. Output:

   • If all pairs have their corresponding reverse pairs (e.g., $(1, 2)$ has $(2, 1)$), the program prints "The relation is symmetric.".
   • If any reverse pair is missing, it prints "The relation is not symmetric.".