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A proposition that is always true is called a *tautology*. A proposition that is always false
is called a *contradiction*. To test if a given proposition is a tautology, a
contradiction or neither we make
for the proposition. If all truth
values in the are T's then the proposition is a tautology, if they
are all F's then it is a contradiction, otherwise it is neither a tautology nor
a contradiction.

Example 1. The proposition p∨∼p is a tautology.

Example 2. The proposition (p⊻q)∧(p↔q) is a contradiction.

Two propositions whose truth tables have the same last column are called * logically
equivalent*. This means that no matter what truth values the primitive
propositions have, these two propositions are either both true or both false. To
test whether or not two propositions are logically equivalent we make a truth
table for each of them and compare their last columns. If they are
then the two propositions are logically equivalent, otherwise they are not.

Example: The proposition (p∧q)∧r is logically equivalent to p∧(q∧r). This is why the expression p∧q∧r is , wherever the parentheses go the result is equivalent.

Similarly, the proposition (p∧q)∧r is logically equivalent to p∧(q∧r) so the expression p∧q∧r is . Wherever the parentheses go the result is equivalent.

*De Morgan's Laws* state that ∼(p∧q) is logically equivalent to ∼p∨∼q
and that ∼(p∨q) is logically equivalent to ∼p∧∼q.

The *converse* of p→q is q→p. The *inverse* of p→q is ∼p→∼q.
The * contrapositive* of p→q is ∼q→∼p.

Example: To find the converse, inverse and contrapositive of the statement "If the train is late then the bus is full" we identify the p: "The train is late" and q: "The bus is full" so that the original statement is of the form p→q.

- The converse is q→p:"If the bus is full then the train is late."
- The inverse is ∼p→∼q:"If the train is not late then the bus is not full."
- The contrapositive is ∼q→∼p: "If the bus is not full then the train is not late."

From the truth tables below we can see that the converse and the inverse are logically equivalent and that the contrapositive is logically equivalent to the original conditional proposition.

The conditional proposition | Its inverse |

Its converse | Its contrapositive |

Example: Consider the proposition "If you are under five feet tall then I am taller than you". It is logically equivalent to its contrapositive, "If I am not taller than you then you are not under five feet tall". Also its converse, "If I am taller than you then you are under five feet tall", is logically equivalent to its inverse, "If you are not under five feet tall then I am not taller than you".