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 X 1, X 2,…, X n  xP(x)  P(x 1 )  P(x 2 )  P(x 3 )  …  P(x n ) 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(x 1 )  P(x 2 )  …  P(x n ) 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)