Start Learning Logic 2 | Disjunction, Tautology and Logical Equivalence [dark version]

Logic 2 adds two core ideas—disjunction and how truth tables enable tautology and logical equivalence—so that complex logical expressions can be simplified safely. Disjunction (“or”) is defined for two logical statements A and B using the symbol ∨. Its truth table outputs true (2) whenever at least one input is true, and it outputs false only when both inputs are false. The transcript also notes a circuit intuition: the two inputs behave like parallel switches, producing a true output if either switch is on.

With disjunction introduced, the next step is using truth tables to analyze compound formulas built from logical operations. A key example is ¬A ∨ A (where negation ¬ binds closest, so parentheses aren’t required). The truth table has two rows—one for A = true and one for A = false—and in both cases the expression evaluates to true. Any formula that always evaluates to true regardless of the truth values of its variables is called a tautology. Tautologies matter because they let a complicated expression be replaced by “true” (2) without changing the logical outcome.

Closely related is logical equivalence, which formalizes when two different formulas can be swapped without changing truth values. Two logical statements are logically equivalent if their truth tables match for every possible assignment of truth values to the variables appearing in them. The transcript emphasizes that this requires checking all combinations of truth values for the shared logical variables and comparing the resulting outputs.

An explicit equivalence example compares (A ∨ B) with ¬A ∧ ¬B. The transcript constructs a truth table over inputs A and B, then computes intermediate results: first A ∨ B, then applies negation to get ¬(A ∨ B); in parallel, it computes ¬A and ¬B and combines them with ∧ to get ¬A ∧ ¬B. For each of the four possible input combinations, the outputs match exactly, establishing that the two expressions are logically equivalent. The notation used for equivalence is a double arrow (↔), indicating that either expression can replace the other inside a larger formula while preserving the overall truth value.

In practical terms, logical equivalence becomes a tool for step-by-step simplification: rather than re-evaluating a complex expression from scratch, one can substitute an equivalent subexpression with a simpler one. The segment closes by signaling that additional logical operations will follow in the next installment, building on the same truth-table methods and substitution rules.