Relations and properties

TheRekz · #1 $Old$ July 10th, 2007, 07:53 PM

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

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

le_su14 · #2 $Old$ July 10th, 2007, 10:10 PM

Quote:

Originally Posted by TheRekz $View Post$

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

TheRekz · #3 $Old$ July 10th, 2007, 10:13 PM

why is $R \cup S$ irreflexive??

le_su14 · #4 $Old$ July 10th, 2007, 10:22 PM

Quote:

Originally Posted by TheRekz $View Post$

why is $R \cup S$ irreflexive??

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

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

Plato · #5 $Old$ July 11th, 2007, 04:53 AM

The set $\Delta _A = \left\{ {(x,x)|x \in A} \right\}$ is known as the diagonal relation on set A. Any relation, $R$ , on A is reflexive if and only if $\Delta _A \subseteq R$ . Using that characterization, it is easy to see the three statements are true.

#6 $Old$ July 12th, 2007, 09:28 PM

Quote:

Originally Posted by le_su14 $View Post$

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

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

Plato · #7 $Old$ July 13th, 2007, 05:01 AM

Quote:

Originally Posted by ali.irfan.kurt $View Post$

Are above statements correct? Is R – S reflexive?

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