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Exercise (Negations Of Types Of Relations)

We have four new definitions above. Negate each of these definitions.

  1. We say that $\mathrm{R}$ is not reflexive iff ...
  2. We say that $\mathrm{R}$ is not symmetric iff ...
  3. We say that $\mathrm{R}$ is not transitive iff ...
  4. We say that $\mathrm{R}$ is not antisymmetric iff ...

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Here are the negations of each of the types of relations.

  1. We say that $\mathrm{R}$ is not reflexive iff there exists $a\in A$ such that $(a,a)\notin \mathrm{R}$.
  2. We say that $\mathrm{R}$ is not symmetric iff there exists $a,b\in A$ such that $(a,b)\in \mathrm{R}$ but $(b,a)\notin \mathrm{R}$. Remember that the negation of the implication "if $(a,b)\in \mathrm{R}$ then $(b,a)\in \mathrm{R}$" is the conjunction "$(a,b)\in \mathrm{R}$ and not $(b,a)\in \mathrm{R}$." The word "but" is logically equivalent to the word "and", yet the word "but" suggests that we expected the opposite to occur but it didn't.
  3. We say that $\mathrm{R}$ is not transitive iff there exists $a,b,c\in A$ such that $(a,b)\in \mathrm{R}$ and $(b,c)\in \mathrm{R}$ but $(a,c)\notin \mathrm{R}$.
  4. We say that $\mathrm{R}$ is not antisymmetric iff there exists $a,b\in A$ such that $(a,b)\in \mathrm{R}$ and $(b,a)\in \mathrm{R}$ but $a\neq b$.