4. (q∨r)↔[(¬q→(r∧¬p))]
Put TTTTFFFF under the p
Put TTFFTTFF under the q's
Put TFTFTFTF under the r's
(q∨r)↔[(¬q→(r∧¬p))]
T T T T T
T F T F T
F T F T T
F F F F T
T T T T F
T F T F F
F T F T F
F F F F F
Under the two ¬'s, put the opposite of what immediately follows the ¬'s and
then erase what follows them:
(q∨r)↔[(¬q→(r∧¬p))]
T T F T F
T F F F F
F T T T F
F F T F F
T T F T T
T F F F T
F T T T T
F F T F T
Under the ∨ put T if there is a T immediately on either side of the ∨,
and an F if there is no T on either side of the ∨. (two F's). Then erase
what was left and right of the ∨.
(q∨r)↔[(¬q→(r∧¬p))]
T F T F
T F F F
T T T F
F T F F
T F T T
T F F T
T T T T
F T F T
Under the ∧ put F if there is an F immediately on either side of the ∧,
and a T if there is no F on either side of the ∧ (two T's). Then erase
what was left and right of the ∧.
(q∨r)↔[(¬q→(r∧¬p))]
T F F
T F F
T T F
F T F
T F T
T F F
T T T
F T F
Under the → put T unless there is a T on the left of the → and an F on
the right of the →. That's the only time you put F under →. Then erase
what is on both sides of the →
(q∨r)↔[(¬q→(r∧¬p))]
T T
T T
T F
F F
T T
T T
T T
F F
Finally, put T under ↔ if what are on both sides of ↔ are alike,
either both T's or both F's, and put F under ↔ if they are different.
Then erase what is on both sides of the ↔
(q∨r)↔[(¬q→(r∧¬p))]
T
T
F
T
T
T
T
T
The answer is TTFTTTTT.
---------------
5. (¬s↔(r→¬q))↔[(s∨p)∧¬(q∧r)]
That will take 16 lines. I'll start it off. You finish using the same rules
as above.
Put TTTTTTTTFFFFFFFF under the p.
Put TTTTFFFFTTTTFFFF under the q's.
Put TTFFTTFFTTFFTTFF under the r's.
Put TFTFTFTFTFTFTFTF under the s's.
(¬s↔(r→¬q))↔[(s∨p)∧¬(q∧r)]
T T T T T T T
F T T F T T T
T F T T T T F
F F T F T T F
T T F T T F T
F T F F T F T
T F F T T F F
F F F F T F F
T T T T F T T
F T T F F T F
T F T T F T F
F F T F F T T
T T F T F F T
F T F F F F T
T F F T F F T
F F F F F F T
Edwin