Disjunction or Alternation
If two statements p and q are combined by the connective “OR” (∨), then the compound statement p ∨ q so formed is called a disjunction. For example, if p: it rains, q: the crops are good, then their disjunction is p ∨ q: it rains or the crops are good.
Sometimes we use the connective “either … or …” to obtain p ∨ q and read p ∨ q as “either p or q”. p ∨ q is false only when both p and q are false. The following truth table shows the truth values of p ∨ q and q ∨ p.
Truth table (p ∨ q, q ∨ p)


p

q

p ∨ q

q ∨ p

T

T

T

T

T

F

T

T

F

T

T

T

F

F

F

F

Rule: p ∨ q is false only when both p and q are false.

Remarks:
 The above truth table shows p ∨ q = q ∨ p.
 The above disjunction is called “inclusive disjunction” as it is true when both p and q are true. In case “p or q” is false when both p and q are true, then it is called “exclusive disjunction” and is written as p ∨ q or p ⊕ q. The truth table for p ∨ q is modified as follows:
Truth table (p ∨ q)


p

q

p ∨ q or p ⊕ q

T

T

F

T

F

T

F

T

T

F

F

F

Rule: p ∨ q is true only when one of pand q is true and the other is false.
