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Solutions to Truth Table Problems

These are the solutions to the truth table exercises. You are strongly advised to work out your own solutions before you look at these.

1. Generate a truth table for the following statements.

a.  A v ~B
Solution 1a
b.  P -> ~Q v R
Solution 1b
c.  ~A v B -> ~C
Solution 1c
d.  (A ^ ~B) -> (C v D)
Solution 1d

2. Convert each statement below  into symbolic form and generate its truth table.

a.  The sun is hot but it is not humid.

P: The sun is hot.
Q: It is humid.
P ^ ~Q

Solution 2a

b.  If John doesn't pass then he will lose his scholarship and drop out of school.

P: John does pass.
Q: John will lose his scholarship.
R: John will drop out of school.
~P -> (Q ^ R)

Solution 2b

c.  If it rains and you don't open your umbrella then you will get wet.

A: It rains.
B: You open your umbrella.
C. You get wet.
(A ^ ~B) -> C

Solution 2c

d. If your car won't start or you don't wake up on time then you will miss your interview and you will not get the new job.

A: Your car will start.
B: You wake up on time.
C: You will miss your interview.
D: You will get the new job.
(~A v ~B) -> (C ^ ~D)

Solution 2d

e. If you elect Mary then Mary will make sure that the federal budget will be balanced, partisan wrangling in Washington will cease, and there will be no cuts in social security benefits.

E: You elect Mary.
B: Mary will make sure that the federal budget will be balanced.
C: Mary will make sure partisan wrangling in Washington will cease.
S: Mary will make sure that there will be no cuts in social security benefits.
E -> B ^ C ^ S
Solution for Part e

f. If the cake gets hot then the icing melts and if the icing melts then the cake cannot be used at the wedding reception.

H: The cake gets hot.
M: The icing melts.
W: The cake can be used at the wedding reception.
(H -> M) ^ (M -> ~W)
Solution to part f

g. If the New York Yankees win the world series or the New York Jets win the super bowl then New York fans will be overjoyed and dance in the streets.

Y: The New York Yankees win the world series.
J: The New York Jets win the super bowl.
O: New York fans will be overjoyed.
D: New York fans will dance in the streets.
(Y v J) -> (O ^ D)
Solution for part g