Asymmetric and Transitive Relations.pptx

The PowerPoint presentation titled *"Asymmetric and Transitive Relations"* explains the concepts of asymmetric and transitive relations in set theory.

- *Asymmetric Relation*: A relation \( R \) on a set \( A \) is asymmetric if, for every \( (x,y) \in R \), the reverse pair \( (y,x) \notin R \). An example with a set \( A = \{1,2,3\} \) and relation \( R = \{(2,1),(2,3),(1,3)\} \) demonstrates asymmetry.
- *Transitive Relation*: A relation \( R \) on a set \( A \) is transitive if, whenever \( (x,y) \in R \) and \( (y,z) \in R \), then \( (x,z) \in R \). An example with \( A = \{1,2,3\} \) and \( R = \{(1,2),(2,3),(1,3)\} \) illustrates transitivity.

1. Asymmetric and Transitive Relations

2. Asymmetric Relation Let A be a set A relation R on set A is said to be asymmetric if ( x,y ) ∈ R then ( y,x ) ∉ R ∀ ( x,y ) ∈ A

3. Example Let A be a set defined as A={1,2,3} R={(2,1),(2,3),(1,3)} The above relation is an asymmetric relation as the relation does not contain any diagonal pair. That is, if ( x,y ) is in R, then ( y,x ) should not be in R.

4. Transitive Relation Let A be a set A relation R on set A is said to be transitive if ( x,y ) ∈ R, ( y,x ) ∈ R then ( x,z ) ∈ R ∀ ( x,y,z ) ∈ A

5. Example Let A be a set defined as A={1,2,3} R={(1,2),(2,3),(1,3)} The above relation is a transitive relation as ( x,y ) exists in R, ( y,z ) exists in R and hence ( x,z ) also should exist in R which is the case.

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