X and Y are equivalent iff X ↔ Y is a tautology
X entails Y  iff X → Y is a tautology

The symbol "∴" is pronounced "therefore" and means "everything that precedes this combines to prove what follows this."

The symbol "⊨" is pronounced "entails" (or "double turnstile") and means "if everything that precedes this is true, what follows must also be true."

The symbol "⊢" is pronounced "proves", "yields", or "turnstile" and means "because I know the things before this, I also know the things after it."


Use
    → inside a logical expression or formula
    ⊨ to express a derivation rule
    ⊢ to mean both ⊨ and "both sides are true"
    ∴ to mean both ⊢ and "and the right-hand side is our goal"