# Relations and properties

• Jul 10th 2007, 05:53 PM
TheRekz
Relations and properties
Suppose that R and S are reflexive relations on a set A. Prove or disprove this statements:
a. $\displaystyle R \cup S$ is reflexive
b. $\displaystyle R \cap S$ is reflexive
c. $\displaystyle R-S$ is irreflexive

for part a and b I answered that it is true and part c is false. is this right?
• Jul 10th 2007, 08:10 PM
le_su14
Quote:

Originally Posted by TheRekz
Suppose that R and S are reflexive relations on a set A. Prove or disprove this statements:
a. $\displaystyle R \cup S$ is reflexive
b. $\displaystyle R \cap S$ is reflexive
c. $\displaystyle R-S$ is irreflexive

for part a and b I answered that it is true and part c is false. is this right?

Hi TheRekz .

I agree with you that : a,b are true and c is false .
• Jul 10th 2007, 08:13 PM
TheRekz
why is $\displaystyle R \cup S$ irreflexive??
• Jul 10th 2007, 08:22 PM
le_su14
Quote:

Originally Posted by TheRekz
why is $\displaystyle R \cup S$ irreflexive??

I'm sorry , I made a mistake .
You are true .

a/ $\displaystyle \forall x \in R \cup S , x \in S$ or $\displaystyle y \in R$. So xRx .$\displaystyle R \cup S$ is reflexive .
b/ $\displaystyle \forall x \in R \cap S , x \in S$ and $\displaystyle y \in R$. So xRx . $\displaystyle R \cap S$ is reflexive .
c/ $\displaystyle \forall x \in R - S , x \in R$ reflexive. So xRx . R - S is reflexive .
• Jul 11th 2007, 02:53 AM
Plato
The set $\displaystyle \Delta _A = \left\{ {(x,x)|x \in A} \right\}$ is known as the diagonal relation on set A. Any relation, $\displaystyle R$, on A is reflexive if and only if $\displaystyle \Delta _A \subseteq R$. Using that characterization, it is easy to see the three statements are true.
• Jul 12th 2007, 07:28 PM
ali.irfan.kurt
Quote:

Originally Posted by le_su14
I'm sorry , I made a mistake .
You are true .

a/ $\displaystyle \forall x \in R \cup S , x \in S$ or $\displaystyle y \in R$. So xRx .$\displaystyle R \cup S$ is reflexive .
b/ $\displaystyle \forall x \in R \cap S , x \in S$ and $\displaystyle y \in R$. So xRx . $\displaystyle R \cap S$ is reflexive .
c/ $\displaystyle \forall x \in R - S , x \in R$ reflexive. So xRx . R - S is reflexive .

Hi guys,

I'm confused. Are above statements correct? Is R – S reflexive? If so how did you come up with the result.

Thanks,
James
• Jul 13th 2007, 03:01 AM
Plato
Quote:

Originally Posted by ali.irfan.kurt
Are above statements correct? Is R – S reflexive?

The relation $\displaystyle R-S$ is irreflexive! Because S is reflexive, the diagonal has been removed.