Truth Table
Mathlet
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This mathlet creates the truth table for a logical compound
proposition. Type one in (followed by the ENTER key) using the following
syntax:
Mathlet Symbol |
Mathematics Symbol |
Meaning |
! |
|
not |
& |
|
and |
| |
|
or |
-> |
|
implies |
<-> |
|
if and only if |
|| |
|
exclusive or |
p |
p |
truth variable |
T |
T |
true |
F |
F |
false |
For a truth variable, any lowercase letter in the ranges a-e, g-s, u-z
(i.e. omitting f and t which are reserved for false and true) may be used. The
negation operator, !, is applied before all others, which are are evaluated
left-to-right. Parentheses, ( ), and brackets, [ ], may be used to enforce a
different evaluation order.
E.g.
p -> q
p & (q | r)
[(p -> q) & (q -> r)] -> (p -> r)
are all syntactically correct.
To Do
The yellow monitor next to an exercise suggests that you should
solve it using the mathlet and a yellow pencil indicates that pencil-and-paper computation
might be necessary.
1. Get familiar with the mathlet by generating the truth tables for the
elementary propositions
and check to see that they're correct!
2. Show that p
q
is logically equivalent to
p
q by creating their truth tables.
3. The converse of p
q is q
p and the
contrapositive of p
q is
q
p. Show that the
converse is
not logically equivalent to the contrapositive by
creating their truth tables.
4. Show that the contrapositive of p
q
is
logically equivalent to p
q by creating their truth tables.
5. Prove the distributive law, p
(q
r) is logically
equivalent to (p
q)
(p
r),
by creating two truth tables.
6. Translate the following sentence into a logical proposition using
three logic variables.
You can connect to the Internet from your dorm room only
if you are a CS major or you have not defaulted on your tuition
payments.
Then use Truth Table to create the truth table for this
proposition. Under what circumstances can a student
not connect to
the Internet from her dorm room? Use the truth table to back up your
assertion. If you are a CS major and have defaulted on your tuition payments,
can you connect? Again, use the truth table to justify your conclusion.
7. Create a logical proposition s in two propositional variables
p and q, with the following truth
table. Use only
,
, and
in your proposition.
p |
q |
s |
T |
T |
F |
T |
F |
T |
F |
T |
F |
F |
F |
T |
Now do the same thing for three propositional variables, p, q, and r.
p |
q |
r |
s |
T |
T |
T |
T |
T |
T |
F |
F |
T |
F |
T |
F |
T |
F |
F |
T |
F |
T |
T |
F |
F |
T |
F |
T |
F |
F |
T |
F |
F |
F |
F |
T |
Informally describe an algorithm to do this for an arbitrary
truth table with n propositional variables.