Mathematics homework help

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Mathematics homework help. Do not submit this page:
Logical Equivalences and Rules of Inference
Propositional Equivalence Name
p q p q    
Definition of Conditional
p T p
p F p
 
 
Identity Laws
p T T
p F F
 
 
Domination Laws
p p p
p p p
 
 
Idempotent Laws
    p p 
Double Negation Law
p q q p
p q q p
  
  
Commutative Laws
   
   
p q r p q r
p q r p q r
    
    
Associative Laws
     
     
p q r p q p r
p q r p q p r
     
     
Distributive Laws
 
 
p q p q
p q p q
     
     
De Morgan’s Laws
 
 
p p q p
p p q p
  
  
Absorption Laws
p p T
p p F
  
  
Negation Laws
p q p q q p         
Definition of Biconditional
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Propositional Rules of Inference Name
p
p q
q



Modus Ponens
q
p q
p




Modus Tollens
p q
q r
p r



 
Hypothetical Syllogism
p q
p
q




Disjunctive Syllogism
p
p q

 
Addition
p q
p



Simplification
p
q
p q

 
Conjunction
p q
p r
q r


 
 
Resolution
Math 11: Final Exam
Name: ____________________________ ID:____________________________________
1. a) Using a truth-table, show that
p q r p q r         
b) Using logical equivalences, show that
p q r p q r         
2. Domain = bit strings (all strings of 0’s and 1’s),
S s s s  : starts with a 0 and ends with a 0
a) Find a Finite State Automata for S.
b) Find a regular expression for S.
3. a) Give a direct proof:
If
n 1
is odd, then
n 1
is odd.
b) Give a direct proof or a proof by contraposition:
If
2
n
is odd, then
n
is odd.
4. Recall a deck of cards has four suits (including hearts) and there are 13 cards in each suit.
a) What is the probability that all the cards in a five card hand will be hearts?
b) What is the probability that all the cards in a five card hand will be the same suit?
c) What is the probability that a five card hand will have exactly three cards of the same suit?
5. Use Djikstra’s Algorithm to find the shortest path from 0 to 6. Show enough work to convince me you
understand the process.
6. a) Find
gcd 259,70  
with E’s algorithm.
b) Use your answer from a) to find integers a and b such that
a b     259 70 7 .
Correct numbers without using your answer above = no credit.
c) Find integers a and b such that
a b     259 70 14
6 d) Solve
25 16mod32 x  . You answer should be positive and less than 32.
e) Find the smallest positive x that satisfies:
x  200 , x 1mod3, x  4mod5
and
x  6mod7
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Mathematics homework help