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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:
  1. The above truth table shows p q = q p.
  2. 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.




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