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:(adsbygoogle = window.adsbygoogle || []).push({});

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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# Homework Help: Relations and Subsets

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