I simply cannot remember which is normalized and which is denormalized. I cannot do coordinate transforms -- I consider myself "spatially impared". And I can never remember the truth table for implies.

"p implies q" or "p only if q" has the following truth table:

     p | q | p -> q
    ---+---+--------
     T | T |   T
     T | F |   F
     F | T |   T
     F | F |   T

where p and q are propositions.

It is logically equivalent to say:

• p implies q
• q is a necessary condition for p
• p is a sufficient condition of q
• in order that p be true it is necessary that q be true
• if p is true then q is true

For example:

    If a person is a father then a person is male.

This statement is of the form p -> q where:

• p: A person is a father
• q: A person is male

It is necessary for a person to be male to be a father. Being a father is a sufficient condition for being male. If a person is not a father, nothing can be said about if they are male. Whereas if a person is not male, they may not be a father. This last statement is the contrapositive of the proposition.

     p | q | -p | -q | p -> q | -q -> -p
    ---+---+----+----+--------+----------
     T | T |  F |  F |   T    |    T
     T | F |  F |  T |   F    |    F
     F | T |  T |  F |   T    |    T
     F | F |  T |  T |   T    |    T

where - represents negation. Since both p -> q and -q -> -p have identical truth tables they are said to be logically equivalent.