1. Discrete Math Relations

Hi,

If a relation is reflexive, symmetric, and transitive; it would mean that the relation is also an equivalence relation? That's my understanding, if I am wrong please correct me.

Say:
R= {(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),( c,c),(d,d),(d,e),(e,d),(e,e),(f,f)}

From this relation I believe that "All except (a,a) , (b,b), (c,c), (d,d), (e,e), (f,f) are equivalence relations"...is that correct or no. I'm not catching on to the whole reflexive, symmetric and transitive part.

The only relation I understand 100% is that transitive is a is the mother of b and b is the mother of c, meaning a is the grandmother of c so it is transitive?

Thanks for any help!

Dan

2. That is correct. An equivalence relation must be reflexive, symmetric and transitive.

Your relation, however, is R. That means R must be reflexive, symmetric and transitive because R represents a relation on {a, b, c, d, e, f}. You do not imply assign the individual pairs as being symmetric etc. It must be the entire relation.

R= {(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),( c,c),(d,d),(d,e),(e,d),(e,e),(f,f)}

A relation is reflexive if it contains all pairs of the form (a,a). Therefore, if R represents a relation on {a,b,c,d,e,f} it must contain (a,a), (b,b), (c,c), (d,d), (e,e) and (f,f), which your relation does.

To test for symmetry, you have to make sure no pair in the relation without the opposite. Meaning is (a,b) is in the relation, you must also have (b,a). Your relation is symmetric.

A relation is transitive if if whenever the relation contains (a,b) and (b,c), it also contains (a,c). This also menas that if it contains the pairs (a,b) and (b,a), it must also have (a,a) and (b,b). Your relation R is therefore also transitive and a equivalence relation.

I hope this helps!

3. Transitive would mean:

If a is b's mom, and b is c's mom, then a is c's mom.

I don't think this is going to hold true.