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
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.