|Sep19-09, 07:09 PM||#1|
Prove or Disprove: A relation ~ on a nonempty set A which is symmetric and transitive must also be reflexive.
If our relation ~ is transitive, then we know: a~b, and b~a [tex]\Rightarrow[/tex] a~a.
Therefore our relation ~ is reflexive, since b~c and c~b [tex]\Rightarrow[/tex] b~b, and c~a and a~c [tex]\Rightarrow[/tex] c~c.
Can the above (idea) constitute a proof in itself?
|Sep19-09, 07:43 PM||#2|
Actually I thought about it a little, and came up with a proof. But can someone critique it and let me know if it's actually alright.
We know ~ is symmetric.
Therefore, [tex]\exists a,b,c \in A[/tex] such that
if a~b, then b~a,But we also know our relation ~ is transitive.
if a~b, and b~a, then a~a, (#1)By (#1), (#2), and (#3) we know our given relation is reflexive.
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