Logical equivalences

Hello, captainjapan!

$(a \to b) \;\equiv\: \sim a \vee b$ . I call it ADI (alternate definition of implication).

Quote:

$(a)\;\;[\sim p \to (q \to r)] \: \equiv\: [q \to (p \vee r)]$

On the left side we have:

. . $\begin{array}{ccc}\sim p \to (q \to r) & & \text{Given} \\ \\ [-4mm] p \vee (q \to r) & & \text{ADI} \\ \\ [-4mm] p \vee (\sim q \vee r) & & \text{ADI} \\ \\ [-4mm] \sim q \vee (p \vee r) & & \text{comm., assoc.} \\ \\[-4mm] q \to (p \vee r) & & \text{ADI} \end{array}$

Quote:

$(b)\;\;\sim (p \to \; \sim q)\: \wedge \sim(p \vee q)$ is a contradiction (i.e. always false).

. . $\begin{array}{ccc} \sim(p \to \:\sim q)\; \wedge \sim(p \vee q) & & \text{Given} \\ \\[-4mm] \sim(\sim p \:\vee \sim q) \;\wedge \sim(p \vee q) & & \text{ADI} \\ \\[-4mm] (p \wedge q) \wedge (\sim p \:\wedge \sim q) & & \text{DeMorgan} \\ \\[-4mm] (p \:\wedge \sim p) \wedge (q \:\wedge \sim q) & & \text{comm, assoc.} \\ f \wedge f \\ \\[-4mm] f \end{array}$