We have shown how to use truth tables to determine if two formulas are truth-functionally equivalent. If two formulas F and G are truth-functionally equivalent we introduce another symbol ↔, aptly called the biconditional. Here is the truth table for the biconditional. p   q    (p ↔ q) 1   1     1 1   0     0 0   1     0 0   0     1 Now we shall say that F and G are truth-functionally equivalent if (F ↔ G) is a tautology. There are other properties of two formulas that we are usually interested in besides truth-functional equivalence. One of these properties is when two formulas are mutually exclusive. We say two formulas F and G are mutually exclusive if (F ∧ G) is contradictory (unsatisfiable). Now using truth tables determine whether the following formulas are truth-functionally equivalent or mutually exclusive. (a) p and ¬p  (b) p and ¬¬p  (c) ¬(p ∧ ¬q) and (p → q)  (d) (¬p ∨ q) and (p → q)  (e) ¬(¬p ∨ ¬q) and (p ∧ q)  (f) ¬(¬p ∧ ¬q) and (p ∨ q)

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