Discrete Mathematics
• Mathematical reasoning: think logically; know
how to prove
• Combinatorial analysis: know how to count
• Discrete structures: represent object and their
relationships
• Algorithmic thinking: how to solve problems by a
compute
• Application and modeling: model application and
solve relevant problems
Chapter 1
Logic , Sets , and Functions
1.1 Logic
– A proposition is a statement that is either true or
false, but not both
– example 1 & 2
– Logical operators (connectives) to form new
proposition
–  P , PQ , PQ , PQ , PQ , PQ
negation , conjunctions, disjunctions , exclusive or , implications , two-way
Logic (cont.)
– Truth table
修過A或B的學生才可以修C
湯或沙拉
the statement “If a player hits more than 60 homeruns,
then a bonus of $10 million is awarded “
in a contract
If today is Friday, then 2+3=5
If today is Friday, then 2+3=6
to search:(MEXICO UNIVERSITEIS)  NEW;
(NEW MEXICO A) UNIV
1.2 Propositional Equivalent
– tautology:
contradiction: (example 1)
contingency:
– logically equivalent: proposition that have
the same truth values in all possible cases
– pq
Propositional Equivalent (cont.)
– P  Q if P  Q is a tautology
P is a tautology if P  T
• example 2
(PQ)PQ
De Margan’s Laws
• example 3 P  Q   P  Q
• example 4 P  ( Q  R )  ( P  Q )  ( P  R )
Propositional Equivalent(cont.)
– Table 5 Logical Equivalence
– example 5&6
• a truth table can be used to determine whether a
compound proposition is a tautology, but only
when a proposition has a small number of
variables
1.3 Predication and Quantifiers
– X > 3 is not a proposition
let P(x) denote the statement X>3, P(4) and
P(2) are propositions
P(x)→predicate,refers to a property X can have
Q(x, y) denote x=y+3, Q(1, 2) is a proposition
Predication and Quantifiers (cont.)
– quantifiers
– xP(x): P(x) is true for all value of X in a
partial domain
– let P(x) denote X+1> X, the domain the set of
real numbers, xP(x) is true
– if the domain contains X1, X2,…, Xn
xP(x) P(x1)  P(x2) P(x3) … P(xn)
Example 8
Predication and Quantifiers (cont.)
– xP(x): P(x) is true for a value of X
• Existential quantifier
– let Q(x) denote X=X+1,xQ(x) is false
– xP(x)  P(x1) P(x2) … P(xn)
Example 11
– xy( x+y=y+x )
x y( x+y=0 )
Predication and Quantifiers (cont.)
Express“some student in this class has visited Mexico”
– M(x): X has visited Mexico, domain of X: students
in the class
 x M(x)
Example “every student in this class has visited either
Canada or Mexico”
– C(x): X has visited Canada
 x ( C(x)  M(x) )
Predication and Quantifiers (cont.)
Example “All lions are fierce”
P(x): X is a lion
Q(x): X is fierce
 x ( P(x)  Q(x) )
“Some lions do not drink coffee”
R(x): X drinks coffee
  x ( P(x)   R(x) )
  x ( P(x)   R(x) )
Predication and Quantifiers (cont.)
Binding variables
– a variable is bound if a quantifiers is used or a
value is assigned ; otherwise, it is free
– a proposition cannot contain free variable
– Q(x,y): X+Y=0
yxQ(x,y) is false
xyQ(x,y) is true
 if yxP(x,y) is true, then xyP(x,y) is true
 if xyP(x,y) is true, then yxP(x,y) is true
Predication and Quantifiers (cont.)
Negations
“Every student in the class has taken a course
in Calculus”
x P(x)
“Some student in the class has not taken a
course in C”
 x  P(x)
 xP(x)  xP(x)
 xQ(x)  x Q(x)