How to Prove the Empty Relation is Transitive, Symmetric but Not Reflexive

relationssolution-verification

Question:

Let $R$ be a relation on a set $A$. Prove that

  1. if $A$ is non-empty, the empty relation is not reflexive on $A$.
  2. the empty relation is symmetric and transitive for every set $A$.

My Solution:

  1. For a relation to be reflexive: For all elements in A, they should be related to themselves.
    ($x$ $R$ $x$). Now in this case there are no elements in the Relation and as A is non-empty no element is related to itself hence the empty relation is not reflexive.

  2. Now for a set to be symmetric and transitive: As these are conditional statements if the antecedent is false the statements would be true. And as the relation is empty in both cases the antecedent is false hence the empty relation is symmetric and transitive.

(I just want to know if my solutions are correct. – I've given brief solutions without using any notation here.)

Best Answer

Your argument for 1 is almost correct. In fact "no element is related to itself" would also hold for $A=\emptyset$, but in that case the empty relation would be reflexive. To make the argument more precise, write something like this: As $A$ is not empty, there exists some element $a\in A$. As $R$ is empty, $a R a$ does not hold, hence $R$ is not reflexive.

Part 2 is absolutely fine.

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