Truth table proofs

Truth tables are used to show all possible values that a given logical expression might take.

Examples

Definition of AND

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

Proof of one of De Morgan's laws

A truth table can also be used to prove a logical identity. The following proves De Morgan'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)	(¬ A) ∨ (¬ B)
false	false	false	true	true
false	true	false	true	true
true	false	false	true	true
true	true	true	false	false

Disproof of a logical fallacy

Truth tables can also be used to disprove a logical fallacy. For example, one fallacy is to assume the converse of an implication holds. If you know that (A → B) is true, and you know that B is true, the fallacy would be to then conclude that A must also be true. Note that the statement - "if an animal is a mammal then it is also a vertebrate" - is true. If we have an animal that is a vertebrate (for example, a dog), it would be a fallacy to now conclude that the animal must also be a mammal.

Consider this truth table.

A	B	A → B (which is equivalent to ¬ A ∨ B)
false	false	true
false	true	true
true	false	false
true	true	true

In the