Necessary and Sufficient Conditions

1. Necessary or Sufficient Condition

S: John is a bachelor. (antecedent) = sufficient condition

N: John is a male. (consequent) = necessary condition

In the set theory notation, S ⊂ N.

Expressions:

If S then N.

S ⇒ N

N if S

S implies N

S only if N

N is implied by S

N when S

N whenever S

S guarantees N

S cannot be true unless N is true.

Derived relations:

If N is false then S is false.

not N ⇒ not S (very important in many applications)

if S is not true, the N can be either true or false.

2. Necessary and Sufficient Condition

P ⇒ Q and Q ⇒ P

P: necessary and sufficient condition for Q

Q: necessary and sufficient condition for P

P and Q are equivalent.

Expressions:

P ⇔ Q

P iff Q

P if and only if Q

Example:

P: Today is the fourth of July.

Q: Today is the independence day in the United States.

Derived relations:

not P ⇔ not Q