Transformational Proof

Transformational proof (↔) proves the equivalence of formulas. It determines that two well-formed formulas of propositional logic, P and Q, are logically equivalent, by transforming $P$ into $Q$ through a sequence of steps that each follow a logical law.

“Transform $P$ into $Q$ to show that $P$ and $Q$ are equivalent”.

Two rules are used implicitly in transformational proofs:

1. Rule of substitution: substituting an equivalent formula for a subformula.
2. Rule of transitivity: If P ↔ Q and Q ↔ R, then P ↔ R. This rule is what connects the steps of the proof.

For Set Theory

There are three styles to use Transformational Proof in Transformational Proof in Set Theory.

First Style

• To prove $A\iff B$, prove $A\leftrightarrow B$

Second Style

• To prove $⊢A=B$, prove $A=B\leftrightarrow true$
  • We can either transform A = B into $\forall x•x\in A\iff x\in B$ (by set // set equality) and proceed from there,
  • or directly transform A and B until they are syntactically the same term (and therefore match the axiom A = A in set theory).

Third Style (my preference)

• To prove $A=B$, prove $x\in A\leftrightarrow x\in B$

I really struggled with this, but I think I am starting to get it.

If you look at your definitions:

Take domain for example, which has the definition $dom(R)=\{a\mid \exists b•(a,b)\in R\}$ You can use it in transformational proof in the following way:

&\leftrightarrow{\exists b \cdot (a,b) \in R}\\ [3pt] \end{aligned}$$ So basically,