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Useful formula

We delve much deeper into propositional (and other kinds of) logic. While we will talk about one more form of logic (first order or predicate logic), we want to get to using our new logical skills on mathematical questions.

As such, let’s end the module on propositional logic with a list of useful and/or well-known provable formulas. Below $A,B,C$ are general formula.

  • \(A \land B \leftrightarrow B \land A\) When we can exchange the placement of an operation taking two inputs (binary operation), then we say it is commutative. This is saying that $\land$ is commutative (up to bi-implication).
  • $\lor$ is commutative : \(A \lor B \leftrightarrow B \lor A\)
  • \((A \land B) \land C \leftrightarrow A \land (B \land C)\) When we have three arguments for a binary and consume them in different orders (apply it to 1 and 2 then the result to 3 vs apply it to 2 and 3 first then the result to 1 while keeping the order of the placements) without affecting the final output, then we say the operation is associative. So here $\land$ is asssociative.
  • You can distribute $\land$ over $\lor$: \(A \land (B \lor C) \leftrightarrow (A \land B) \lor (A \land C)\)
  • You can also distribute $\lor$ over $\land$: \(A \lor (B \land C) \leftrightarrow (A \lor B) \land (A \lor C)\)
  • \[(A \to (B \to C)) \leftrightarrow (A \land B \to C)\]
  • Transitivity of implication: \((A \to B) \to ((B \to C) \to (A \to C))\)
  • \[((A \lor B) \to C) \leftrightarrow (A \to C) \lor (B \to C)\]
  • \[\neg (A \lor B) \leftrightarrow \neg A \land \neg B\]
  • \(\neg (A \land B) \leftrightarrow \neg A \lor \neg B\)
  • \[\neg (A \to B) \leftrightarrow A \land \neg B\]
  • \[\neg A \to (A \to B)\]
  • \((\neg A \lor B) \leftrightarrow (A \to B)\)
  • \[\neg(A \leftrightarrow \neg A)\]
  • \[(A \to B) \leftrightarrow (\neg B \to \neg A)\]
  • \((A \to (B \lor C)) \to ((A \to B) \lor (A \to C))\)
  • If $B \leftrightarrow C$, then for any $A$ we have
    • \[A \lor B \leftrightarrow A \lor C\]
    • \[A \land B \leftrightarrow A \land C\]
    • \[A \to B \leftrightarrow A \to C\]

Each of the above can provide a useful logical shortcut when in the midst of a mathematical argument.