- #1
tn0c
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Hi, this is my first time posting here, and I am trying to prove the following proofs and I do not know how to start:
Suppose R and S are relations on set A
1. If R is reflexive and R is a subset of S, then S is reflexive.
2. If R is symmetric and R is a subset of S, then S is symmetric.
3. If R is transitive and R is a subset of S, then S is transitive.
I understand the definitions well enough, but I do not know how to specify these conditions.
I am not certain that any of these are true. Any help would certainly be appreciated.
Attempt on #1:
Assume R is reflexive
Assume R is a subset of S.
Since R is reflexive, for all x in R, xRx ---> xRx
and since R is a subset of S, all x in R is in S as well.
Thus x in S, xSx ---> xSx (this is the step I am not sure of, nor do I know if I have anything else to work off of.)
Therefore S is reflexive. The other two questions I set up the same way, but I use the other definitions. It definitely is not enough, and in the original problem it states (prove or show counterexample). So I am wondering if this is suitable.
1. let R, S be relations on A
R= { (1,1) (2,2) (1,2) (2,1)}
S= { (1,1) (2,2) (1,2) (2,1) (1,3) (2,3) (3,2) (3,1)}
Clearly R is reflexive and R is in S. But S is not reflexive as it does not contain (3,3).
So for reflexivity at least this does not seem to be true. I am however trying to figure out if that is the case for symmetric and transitive.
Suppose R and S are relations on set A
1. If R is reflexive and R is a subset of S, then S is reflexive.
2. If R is symmetric and R is a subset of S, then S is symmetric.
3. If R is transitive and R is a subset of S, then S is transitive.
I understand the definitions well enough, but I do not know how to specify these conditions.
I am not certain that any of these are true. Any help would certainly be appreciated.
Attempt on #1:
Assume R is reflexive
Assume R is a subset of S.
Since R is reflexive, for all x in R, xRx ---> xRx
and since R is a subset of S, all x in R is in S as well.
Thus x in S, xSx ---> xSx (this is the step I am not sure of, nor do I know if I have anything else to work off of.)
Therefore S is reflexive. The other two questions I set up the same way, but I use the other definitions. It definitely is not enough, and in the original problem it states (prove or show counterexample). So I am wondering if this is suitable.
1. let R, S be relations on A
R= { (1,1) (2,2) (1,2) (2,1)}
S= { (1,1) (2,2) (1,2) (2,1) (1,3) (2,3) (3,2) (3,1)}
Clearly R is reflexive and R is in S. But S is not reflexive as it does not contain (3,3).
So for reflexivity at least this does not seem to be true. I am however trying to figure out if that is the case for symmetric and transitive.
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