I'm having trouble understanding the concepts of relational proofs. I understand the definition on relation, but I'm having trouble understanding the proof process of the reflexive, symmetric, and transitive properties. Check if the relation R is a equivalence relation.
$ R = \{(x,x): x \in Z\}$
Proof: Let $a\in Z$. then $ \forall a, \exists a : a = a$ Therefore $(a,a) \in R$ since R is all possible ordered pairs with the same entry, so R is reflexive.
Proof: Let $b,c\in Z$. Then $\forall (b,c) \in Z^2, \exists (c,b) \in Z^2$. Also since $(a,b) \in R$ when a=b, then $(b,a) \in R $ when b=a, therefore aRb $\implies$ bRa. Also So since ∀b,c∈Z,(b,c)∉R ∀b,c∈Z,(b,c)∉R where b!=c so the antecedent of the implication will be false, therefore the implication will be true. Which covers all cases for a,b $\therefore$ R is symmetric.
Does this look correct for the symmetric proof. I feel really uncertain about it. Note: I left out the transitive proof bc I want to understand the other parts before I try that. Thank you.
You need to be more precise and more direct in your symmetry proof:
Start by picking an arbitrary $(a,b) \in R$. This means by definition that $(a,b)=(x,x)$ for some $x \in \Bbb Z$. Ergo $a=x$ and $b=x$ and so $a=b$ and then $(b,a)=(x,x) \in R$ as well, and you're done. (For an arbitrary $(a,b) \in R$ we have shown that also $(b,a) \in R$.)