- #1
jeff1evesque
- 312
- 0
Statement:
Prove or Disprove: A relation ~ on a nonempty set A which is symmetric and transitive must also be reflexive.
Ideas:
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.
Proof:
Can the above (idea) constitute a proof in itself?
Thanks,
Jeffrey
Prove or Disprove: A relation ~ on a nonempty set A which is symmetric and transitive must also be reflexive.
Ideas:
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.
Proof:
Can the above (idea) constitute a proof in itself?
Thanks,
Jeffrey