We are asked to complete the following truth table.
p
q
~p
~p∨ q
F
T
T
T
F
F
T
F
Let's fill the cells by column.
Completing the Column of ~p
We can read ~p as notp; the statement ~p is the negation of p. These two statements have opposite truth values. To fill this column, we do not need the information in column q.
p
q
~p
~p∨ q
F
T
T
T
T
F
F
F
T
T
F
F
Completing the Column of ~p∨ q
We can read ~p ∨q as ~p or q.
The statement ~p ∨ q is the disjuction of ~p and q. It is true when either ~p or q is true, so we write T if there is a T in the same row in either the ~p or q columns.
To fill this column we do not need to look at the truth value of p, because we must use the truth values of ~p we already found.
p
q
~p
~p∨ q
F
T
T
T
T
T
F
T
F
F
T
T
T
F
F
F
Completed Truth Table
The completed truth table is as follows.
p
q
~p
~p∨ q
F
T
T
T
T
T
F
T
F
F
T
T
T
F
F
F
Extra
Order of Operation
It is important to remember, that in ~p∨ q we are taking the disjunction of q and the negation of p.
The negation comes first, so ~p∨ q=( ~p)∨ q.
This expression is not the same as ~(p∨ q).
