Common Mistakes on HW 8

• R \subseteq A \times A is antisymmetric (on A) iff \forall (a,b) \in R, a \ne b \rightarrow (b,a) ∉ R – the condition a \ne b was often left out in the proof of #21(d).

• \emptyset \subset A × A is antisymmetric – no 2 elements in the empty set contradicts antisymmetry. The '\emptyset' relation is also (vacuously) symmetric, transitive, but is not necessarily reflexive. \emptyset is reflexive on A only if A = \emptyset.

• R = \{(1,2),(2,3),(1,3),(2,1),(3,2)\} is not a transitive relation on \{1,2,3\} because (1,2),(2,1) \in R, but (1,1) ∉ R.

• Other problems with proof writing: a proof has to be written in complete sentence with necessary conjunctions such as “suppose”, “if”, “then”, “else”, “such that”, etc, but everything else should be in mathematical notations.
• In particular, use

  • "let (assumption) be true" to state assumptions,

  • "therefore/thus (conclusion)" to state things that follow from your assumptions, and

  • "To show: (claim)" to state or restate (equivalently) what you are trying to show.