Question

In a set of integers, is the Relation R={(1,2),(2,1)} symmetrical, transitive or reflexive?

Answer 1

■ Set of Integers :

  The set of integers is computed with the whole numbers 0, ± 1, ± 2, ± 3, ... and is denoted by ℤ, and the set of Natural Numbers ℕ is the set of all positive integers ℤ⁺.

■ Set Relations ( ρ ) :

  Let us consider S to be a non - empty set and we take a binary relation ρ on it.

• Reflexive relation --- when for any element a in S, we will find a relation

  (a, a) ∈ ρ ,

the relation ρ is said to be Reflexive.

• Symmetric relation --- when for any two elements a, b in S, we will find a relation

  (a, b) ∈ ρ ⇒ (b, a) ∈ ρ ,

the relation ρ is said to be Symmetric.

• Transitive relation --- when for any three elements a, b, c in S, we will find a relation

  (a, b) ∈ ρ and (b, c) ∈ ρ ⇒ (a, c) ∈ ρ ,

the relation  is said to be Transitive.

• Equivalence relation --- when any relation ρ on S be Reflexive, Symmetric and Transitive at a time, the relation is called Equivalence.

■ Solution of the given problem :

The given relation is

  R = { (1, 2), (2,1) }

• Checking for Reflexive relation -

  There are two elements 1 and 2 in ℤ. But there is no (1, 1) or (2, 2) such that (1, 1) or (2, 2) belongs to R. Thus, R isn't Reflexive.

• Checking for Symmetric relation -

  We see that for the two elements 1 and 2 in ℤ, there is a relation

 (1, 2) ∈ R ⇒ (2, 1) ∈ R ,

and so R is Symmetric.

• Checking for Transitive relation -

  There is only two elements 1 and 2 in ℤ. So to satisfy the Transitive relation, the third element is missing. Thus, R isn't Transitive.

• Checking for Equivalence relation -

  Since R isn't Reflexive, Symmetric and Transitive at a time, R isn't an Equivalence relation.

❈ Therefore, in set of integers ℤ, the relation R = { (1, 2) , (2, 1) } is Symmetric only.