Definition Of Disjunction And its Uses

A joint statement asserting that at least one of the constituent statements is true, so that more then one constituent statements can also be true, certainly all are not false is compounded by the use of ‘either…… or’ or simply OR. A compound statement of two statements will be true if either of them is true or both are true. It p stands for the optional Maths, and q for the optional Statistics, p OR q will signify;

P and q   Maths and Statistics
P and ~q Maths and not Statistics
Q and ~p Statistics and not Maths

This relationship will be brought out more clearly by the following truth table which may be compared with the previous one to grasp the difference.

As shown above p or q is false only when both p and q are false. This obviously is an inclusive type of disjunction which covers also the situation when both are true.
The plain circuit depicting the relationship is given here under:

The electrical impulse from initial point S_1 will not pass to the terminal point S_2 only when both the switches p and q are open (i.e. off)

The exclusive type of disjunction, which keeps out the situation when all the constituent statement are true, is indicated by p OR q or p+q. In this case the first row of the truth table 4 will have F in the output column. Its circuit will be of the following type :

The exclusive type of disjunction which conveys the sense of “p or q but not both“ is a special case of OR connective. Therefore, unless otherwise stated, p OR q will be taken in the inclusive sense which will always mean “p OR q or both” or “p and/or q”