Math 2534
Homework sheet 2
Spring 2012
Problem 1: Use Algebra of logic to prove the following equivalences. You must justify each step of the
proof. (Don’t forget the double negative law :>)
A)Theorem:
[( p  q)
p]  q  T
Proof:
[( p  q)
p]  q 
[( p  q)
[( p 
p]  q 
p)  (q 
[( F )  (q 
[(q 
[ q
given
Equivalent form of implication
p)]  q 
p)]  q 
Distrubution Law
Inverse Law
p)]  q 
Identity
p]  q 
DeMorgan's Law
[ q  p]  q 
Double negative
( q  q)  p 
Commutative Law and Associative Law
Tp
Inverse Law
T
Universal Bound Law
Therefore [( p  q)
p]  q  T
B) Theorem:
[ p  (q  r )]  ( q  p )  p  ( q  r )
Proof:
[ p  (q  r )]  ( q  p ) 
Given
[ p  ( q  r )]  ( q  p) 
Equivalent form of implication
[ p  ( q  r )]  ( p 
Commutative Law
p  [( q  r )
p  [( q
q] 
q)  r ] 
p  [( q)  r ] 
q) 
Distributive Law
Commutative and Associative Laws
Idempotent Law
Therefore: [ p  (q  r )]  ( q  p )  p  ( q  r )
Problem 2: Put the following statements into symbolic implication form and define all the variables.
Indicate which is the sufficient condition and which is the necessary condition in each statement.
1) You will find the treasure if you find the map.
T : “you will find the treasure” is the necessary condition,
M: “you find the map” is the sufficient condition
M T
2) The treasure is yours only if you find the map.
T : “you will find the treasure” is the sufficient condition,
M: “you find the map” is the necessary condition
T M
3) You take the medicine or you are not sick.
M: You take the medicine
S: you are sick
M
S  M  S S M
This is the equivalent form of implication and contrapositive form.
S is the sufficient condition and M is the necessary condition.
Problem 3: Are the following two statement equivalent? Indicate the sufficient and necessary
conditions. Put into symbolic implication and justify your conclusion.
1) The dinner will be a success only if John cooks.
2) John did cook or the dinner did not go well.
Solution: D: Dinner is a success and J: John cooks
1) D  J
2) J
D is the sufficient condition and J is the necessary condition
D  J  D  D  J D is the sufficient condition and J is the necessary condition
Since the sufficient and necessary conditions are the same for 1) and 2) they are equivalent statments.
Problem 4: Are the following arguments valid? Justify you conclusion.
a) If you do not attend the meeting, you will not be promoted.
You are promoted
Therefore you attended the meeting.
Let M be the statement “you attend the meeting” and P will be the statement “ you will be promoted”
The argument is as follows:
M P
P
M
This is a valid argument since we have the contrapositive statement: P  M
b) If you finish your chores we can go to the game.
We go to the game.
Therefore you finished your chores.
Let C be the statement “you finish your chores” and G be the statement “we go to the game”.
The argument is as follows:
C G
G
C
This is not a valid argument since G is the necessary condition and cannot guarantee the
sufficient condition.
Problem 5: A Logic Puzzle and Working with Hypothesis
For your solution to the logic puzzle below, you need to put all statements into symbolic logic
and define your variables. Then write a concise paragraph that justifies your reasoning and
your conclusion. Indicate clearly the sufficient and necessary conditions.
Report Card
Three siblings Alice, Bob and Carol truthfully reported their grades to their parents as follows:
Alice: If I passed math, then so did Bob.
I passed English if and only if Carol did.
Bob: I passed math only if Alice did.
Alice did not pass History.
Carol: Either Alice passed history or I did not pass it.
If Bob did not pass English, then neither did Alice.
If each of the three passed at least one subject and each subject was passed by at least one of
the three, and if Carol did not pass the same number of subjects as either of her siblings, which
subjects did they each pass?
Define your variables as follows:
A E means that Alice passed English
A M means that Alice passed Math
A H means that Alice passed History
BE means that Bob passed English
BM means that Bob passed Math
BH means that Bob passed History
CE means that Carol passed English
CM means that Carol passed Math
CH means that Carol passed History
Symbolic Logic:
AM  BM
AE  CE
BM  AM
AH
AH 
CH
BE  AE
We know that Alice did not pass history so the statement that Alice passed history
is false. In order for the exclusive statement AH  CH to be true CH must be
true. Therefore we now know that CH is false and Carol did not pass history.
Since we a given that a least one person must pass each subject, we know that Bob
passed history.
Next we notice that
BE  AE  AE  BE by contrapositive. We also have the
biconditional AE  CE So CE  AE  BE and we have that they all passed
English or they all did not pass English. Again, we know that at least one person
had to pass each subject so they all passed English.
We also notice that we have AM  BM and BM  AM , which is also biconditional
and AM  BM . This means that both Alice and Bob passed Math or both did not.
We know that everyone has passed English. We also know that Bob passed
History. This means that Bob passed two courses. The only subject left to
consider is Math. If Bob and Alice did not pass Math then Carol had to pass
Math. This would mean that Bob and Carol passed 2 courses and Alice passed
one. But we know that Carol did not pass the same number of courses as either
Alice or Bob. So Bob and Alice must have passed Math. This will give us the
following information:
Carol passed English
Bob passed English, History and Math.
Alice passed English and Math.