We are asked to complete the following truth table.
p
q
~q
~q∧ p
F
F
T
T
T
F
F
T
Let's fill the cells by column.
Completing the Column of q
We can read ~q as notq. The statement ~q is the negation of q. These two statements have opposite truth values. To fill in this column we do not need the information in column p.
p
q
~q
~q∧ p
F
T
F
T
F
T
T
T
F
F
F
T
Completing the Column of ~q∧ p
We can read ~q ∧p as ~q and p.
The statement ~q ∧ p is the conjuction of ~q and p. It is only true when both ~q and p are true, so we write T only when the entries in the same row in the ~q and p column are both T. Otherwise we write F.
To fill in this column we do not need to look at the truth value of q.
p
q
~q
~q∧ p
F
T
F
F
T
F
T
T
T
T
F
F
F
F
T
F
Completed Truth Table
The completed truth table is as follows.
p
q
~q
~q∧ p
F
T
F
F
T
F
T
T
T
T
F
F
F
F
T
F
Extra
Order of Operation
It is important to remember, that in ~q∧ p we are taking the conjunction of p and the negation of q.
The negation comes first, so ~q∧ p=( ~q)∧ p.
This expression is not the same as ~(q∧ p).
