Kinds Of Function ( Reflexive, Symmetric , Transitive Antisymmetric ) By Ishrat Hayat Malik

Kinds Of Function ( Reflexive, Symmetric , Transitive Antisymmetric ) By Ishrat Hayat Malik

Kinds Of Function In Discrete Structure

1) Reflexive
2) Symmetric
3) Transitive
4) Anti-Symmetric

• Reflexive: The relation R on {1,2,3} given by

R = {(1,1), (2,2), (2,3), (3,3)} is reflexive. (All
loops are present.)

• Symmetric: The relation R on {1,2,3} given by

R = {(1,1), (1,2), (2,1), (1,3), (3,1)} is symmetric.
(All paths are 2-way.)

• Transitive: The relation R on {1,2,3} given by

R = {(1,1), (1,2), (2,1), (2,2), (2,3), (1,3)} is
transitive. (If I can get from one point to another
in 2 steps, then I can get there in 1 step.)

Anti-Symmetric  

R = {(1,1), (1,2), (3,2), (3,3)} is antisymmetric. • Is every relation symmetric or anti-symmetric? • No! Consider R = {(1,2), (2,1), (1,3)}

Violations of the Properties

• Why is R = {(1,1), (2,2), (3,3)} not reflexive on
{1,2,3,4}?
Because (4,4) is missing.
• Why is R = {(1,2), (2,1), (3,1)} not symmetric?
Because (1,3) is missing.
• Why is R = {(1,2), (2,3), (1,3), (2,1)} not
transitive?
Because (1,1) and (2,2) are missing.
• Is {(1,1), (2,2), (3,3)} symmetric? transitive?
 Yes! Yes!