Chapter 2 - Reasoning Using Equivalence Transformations

<aside> 💡 Two propositions are equivalent if they have the same value in every state.

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The Laws of Equivalence

Definition

Propositions $E1$ and $E2$ are equivalent if and only if $E1 = E2$ is a tautology.

In other words, an equivalence is an equality that is a tautology.

• Laws of Equivalence

The Rules of Sustitution and Transitivity

Rule of Substitution

Let $e1 = e2$ be an equivalence and $E(p)$ be a proposition, written as a function of one of its identifiers $p$. Then $E(e1) = E(e2)$ and $E(e2) = E(e1)$ are also equivalences.

Rule of Transitivity

If $e1 = e2$ and $e2 = e3$ are equivalences, then so is $e1 = e3$.

A Formal System of Axioms and Inference Rules

Axioms

Any proposition that arises by substituting propositions for $E1$, $E2$ and $E3$ in one of the Laws 1-12 is called a theorem.