# Reflexivity Implies Symmetry?

1. May 20, 2014

### knowLittle

1. The problem statement, all variables and given/known data
Is this relation, R, on $S= \{ 1, 2, 3 \} \\ R = \{ (1,1), (2,2) , (3,3) \}$

Symmetric?
It is obvious that it is reflexive.

2. May 20, 2014

### knowLittle

Nevermind. I just read somewhere that reflexive statements don't count towards symmetry. Apparently, it involves something like a diagonal class; I guess they pair this combinations in a matrix like form.

Anyway. Thanks.

3. May 20, 2014

### HallsofIvy

Staff Emeritus
The relation R= {(1, 1), (2, 2), (3, 3), (1, 3)} is "reflexive" but not "symmetric" so reflexive does not "imply" symmetry. However, in this case there is no (x, y) in the relation without a corresponding (y, x) so this particular example is both reflexive and symmetric.

4. May 20, 2014

### pasmith

The relation R can be described as "xRy if and only if x = y". Thus R is an equivalence relation because equality is an equivalence relation. Hence R is reflexive, symmetric and transitive.

5. May 20, 2014

### knowLittle

Wait, so my R is an equivalence relation then? Supposedly, it partitions the set into disjoint classes. I guess that my classes would be [1], [2], [3]?

HallsofIvy, thank you for the clarification. I should have stated that in this case, it means the same.

6. May 20, 2014

### micromass

Staff Emeritus
Yes.