p q ~p ~q q⋀~p
T T F F ?
T F F T ?
F T T F ?
F F T T ?
You need to learn the rules for the four basic
symbols ⋀, ⋁, ->, and <->
Rules:
⋀ is T only for T⋀T, otherwise it's F.
⋁ is F only for F⋁F, otherwise it's T.
-> is F only for T->F, otherwise it's T.
<-> is T for T<->T, F<->F. <-> is F for T<->F, F<->T.
You only have ⋀ here to do, but you need to learn
those other three rules as well for other truth tables.
Here goes:
p q ~p ~q q⋀~p
T T F F ?
T F F T ?
F T T F ?
F F T T ?
Let's look at q⋀~p.
Let's do the top line:
On the left of the ⋀ in q⋀~p is q, and on the top line q has T under it.
On the right of the ⋀ in q⋀~p is ~p, and on the top line ~p has F under it.
So on the top line q⋀~p is a case of T⋀F. So we look at the rule
for ⋀, which is:
"⋀ is T only for T⋀T, otherwise it's F."
T⋀F is not a case of T⋀T, and therefore it's F. So we put an F
on the top row underneath q⋀~p. So we now have this:
p q ~p ~q q⋀~p
T T F F F
T F F T ?
F T T F ?
F F T T ?
--------------------------------------------------------------
Next let's do the second line:
On the left of the ⋀ in q⋀~p is q, and on the 2nd line q has F under it.
On the right of the ⋀ in q⋀~p is ~p, and on the 2nd line ~p has F under it.
So on the 2nd line q⋀~p is a case of F⋀F. So we look at the rule
for ⋀, which is:
"⋀ is T only for T⋀T, otherwise it's F."
F⋀F is not a case of T⋀T, and therefore it's F. So we put an F
on the 2nd line underneath q⋀~p. So we now have this:
p q ~p ~q q⋀~p
T T F F F
T F F T F
F T T F ?
F F T T ?
--------------------------------------------------------------
Next let's do the third line:
On the left of the ⋀ in q⋀~p is q, and on the 3rd line q has T under it.
On the right of the ⋀ in q⋀~p is ~p, and on the 3rd line ~p has T under it.
So on the 3rd line q⋀~p is a case of T⋀T. So we look at the rule
for ⋀, which is:
"⋀ is T only for T⋀T, otherwise it's F."
T⋀T is INDEED a case of T⋀T, and therefore it's T. So we put a T
on the 3rd row underneath q⋀~p. So we now have this:
p q ~p ~q q⋀~p
T T F F F
T F F T F
F T T F T
F F T T ?
--------------------------------------------------------------
Next let's do the bottom line:
On the left of the ⋀ in q⋀~p is q, and on the bottom line q has F under it.
On the right of the ⋀ in q⋀~p is ~p, and on the bottom line ~p has T under it.
So on the bottom line q⋀~p is a case of F⋀T. So we look at the rule
for ⋀, which is:
"⋀ is T only for T⋀T, otherwise it's F."
F⋀T is not a case of T⋀T, and therefore it's F. So we put an F
on the 2nd row underneath q⋀~p. So finally we have this:
p q ~p ~q q⋀~p
T T F F F
T F F T F
F T T F T
F F T T F
Do you now see how truth tables are done? You have to learn the
rules for all four symbols ⋀, ⋁, ->, and <-> to do truth tables.
Edwin
