BCA PART A 5.R program to check whether the given relation is reflexive.

 

In mathematics, a relation $R$ on a set $A$ is called reflexive if every element in the set $A$ is related to itself. In other words, for a relation to be reflexive, the pair $(a,a)$ must be in the relation $R$ for every element $a$ in $A$.

Formal Definition:

A relation $R$ on a set $A$ is reflexive if: $\mathrm{\forall }a\in A,(a,a)\in R$ This means that each element $a$ in the set $A$ must appear in a pair where it is related to itself.

Example:

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

In this case, $R$ is reflexive because every element of $A$ (1, 2, 3) has a corresponding pair in $R$ where it is related to itself (i.e., $(1,1),(2,2),(3,3$

Non-Reflexive Example:

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

Here, $R$ is not reflexive because the element 3 in set $A$ does not have a pair $(3,3)$ in the relation $R$.

Applications:

Reflexive relations are important in various fields of mathematics, such as:

• Equivalence Relations: Reflexivity is one of the three properties (along with symmetry and transitivity) that define an equivalence relation.
• Order Relations: Reflexive relations are also used in the context of orderings, like partial or total orders, where each element is considered to be at least equal to itself.

Program:

# Function to check if a relation is reflexive

is_reflexive <- function(relation, setA) {

  # Iterate over each element in setA

  for (a in setA) {

    # Check if the pair (a, a) exists in the relation

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

      return(FALSE)

    }

  }

  return(TRUE)

}

# Example usage

# Define the set A

setA <- c(1, 2, 3)

# Define the relation R as a matrix of pairs

# Each row represents a pair (x, y)

relation <- matrix(c(1, 1,

                     2, 2,

                     3, 3), 

                   ncol = 2, byrow = TRUE)

# Check if the relation is reflexive

if (is_reflexive(relation, setA)) {

  print("The relation is reflexive.")

} else {

  print("The relation is not reflexive.")

}

Output:

[1] "The relation is reflexive."

Explanation:

1. Function is_reflexive:

   • Input:
     • relation: A matrix representing the relation $R$. Each row in the matrix is a pair $(x,y).$
     • setA: The set $A$ on which the relation is defined.
   • Logic:
     • The function loops through each element $a$ in setA.
     • For each element $a$, it checks if the pair $(a,a)exists\; in\; the\; relation.$
     • If any element does not have the pair $(a, a)$ in the relation, the function returns FALSE, indicating the relation is not reflexive.
     • If all required pairs are present, it returns TRUE, indicating the relation is reflexive.

2. Example Usage:

   • Set $A$: Defined as setA <- c(1, 2, 3).
   • Relation $R$: Defined as a matrix with pairs relation <- matrix(c(1, 1, 2, 2, 3, 3), ncol = 2, byrow = TRUE).
   • The program checks if each element in setA is related to itself (i.e., if $(a, a)$ exists for each $a$ in setA).

3. Output:

   • If the relation is reflexive, it prints "The relation is reflexive.".
   • If the relation is not reflexive, it prints "The relation is not reflexive.".