Relations: R1 = {(1,1), (1,2), (2,1), (2,2), (3,4), (4,1), (

[solomon_13000]

Given X = {1,2,3,4}, which of the following relations are transitive, reflexive, symmetric, antisymmetric, partial order, equivalence relation.

R1 = {(1,1), (1,2), (2,1), (2,2), (3,4), (4,1), (4,4)}

solution:

Reflexive - Yes

transitive - No because for (3,4)(4,1) = (3,1) cannot be found in the relation

symmetric - Yes

Antisymmetric - No because the elements can be found in the relation

Is this correct?

 

[pka]

Do you know the definitions?
I appears as if you do not.
You have only one of the four correct.
Please reply with the complete definitions.

 

[solomon_13000]

Reflexive - for every element there must be a mirror image
Symetric - if (x,y) E R then (y,x} E R
Transitive - if (x,y) E R and (y,z) E R then (x,z) must exist

 

[solomon_13000]

which is correct?

 

[skeeter]

Re: Sets - Relations

Given X = {1,2,3,4}, which of the following relations are transitive, reflexive, symmetric, antisymmetric, partial order, equivalence relation.

R1 = {(1,1), (1,2), (2,1), (2,2), (3,4), (4,1), (4,4)}

solution:

Reflexive - Yes where is (3,3)?

transitive - No because for (3,4)(4,1) = (3,1) cannot be found in the relation

symmetric - Yes I see (3,4) ... where is (4,3)?

Antisymmetric - No because the elements can be found in the relation
can't help with this one ... don't know the definition

Is this correct?

 

[solomon_13000]

Antisymmetric is opposite of symmetric.

 

[pka]

Antisymmetric is opposite of symmetric.

No indeed it is not!
Antisymmetric: \(\displaystyle \left( {{\rm{a,b}}} \right) \in R\;\& \;\left( {b,a} \right) \in R \Rightarrow \;a = b.\)