Truth table proofs

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Truth tables are used to show all possible values that a given logical expression might take. For example, the following gives the definition of the logical AND operation.

A B A ∧ B
false false false
false true false
true false false
true true true

A truth table can also be used to prove a logical identity. The following proves DeMorgan's law that ¬ (A ∧ B) is equivalent to (¬ A) ∨ (¬ B). Notice that the truth values for these (the last two columns in the table) are always the same.

A B A ∧ B ¬ (A ∧ B) (&not A) ∨ (&not B)
false false false true true
false true false true true
true false false true true
true true true false false