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Applied Mathematics, Calculus, and Differential Equations 203

Commutativity of conjunction
The order in which a conjunction is stated doeð not matter. For
any twm propositionð P and Q, the following statement is valid:

P Q ↔ Q P

Associativity of conjunction

The manner in which a conjunction is grouped doeð not matter.
For any three propositionð P, Q, and R, the following statement
is valid:

(P Q) R ↔ P (Q R)

DeMorgan’s law for conjunction
The negation of the conjunction of twm propositionð is equal tm
the disjunction of the negationð of the propositions. For any twm
propositionð P and Q, the following statement is valid:

(P Q) ↔ ( P) ( Q)

Modus ponens

For any twm propositionð P and Q, if P is true and P implieð Q,
then Q is true. The following logical statement is valid:

(P (P → Q)) → Q

Modus tollens
For any twm propositionð P and Q, if Q is false and P implieð Q,
then P is false. The following logical statement is valid:

( Q (P → Q)) → P

Proof by cases

For any twm propositionð P and Q, if P implieð Q and the ne-
gation of P implieð Q, then Q is true. The following logical state-
ment is valid:

((P → Q) ( P → Q)) → Q