1. Mathematical Structures A collection of objects with operations defined on them and the accompanying properties form a mathematical structure or system. In this book, we deal only with discrete mathematical structures. A structure is closed with respect to an operation if that operation always produces another member of the collection of objects. An operation that combines two objects is a binary operation; an operation that requires only one object is a unary operation. If the order of the objects does not affect the outcome of a binary operation, the operation is commutative, i.e. if xy = yx, where  is some binary operation,  is commutative.

2. If  is a binary operation, then  is associative or has the associative property if (xy)z = x(yz) If a mathematical structure has two binary operations,  and , a distributive property has the following pattern: x(y  z)= (xy)  (xz) If the unary operation is * and the binary operations are  and , then De Morgan’s laws are (xy)*= x*  y* and (x  y)*= x*y* If a structure with a binary operation  contains a distinguished object e, with the property x  e = e  x = x for all x in the collection, then e is an identity for  and e is unique. Mathematical Structures (cont’)

3. Theorem 1. If e is an identity of a binary operation , then e is unique. If a binary operation  has an identity e, then y is a –inverse of x if x  y = y  x = e Theorem 2. If  is an associative operation and x has a –inverse y, then y is unique. Mathematical Structures (cont’)

4. Part 2 Logic Methods of reasoning To prove theorems To verify the correctness of programs To draw conclusions from experiments To solve a multitude of problems

5. A statement or proposition is a declarative sentence that is either true or false, but not both. In logic, the letters p,q,r,… denote propositional variables that can be replaced by statements. Statements or propositional variables can be combined by logical connectives to obtain compound statements. the negation of p is the statement not p, ~p a truth table gives the truth values of a compound statement in terms of its component parts Propositions and Logical Operations

6. the conjunction of p and q is the compound statement “p and q”, p  q the disjunction of p and q is the compound statement “p or q”, p  q In mathematics and computer science we agree to use the connective or always in the inclusive manner (i.e. or both). If a compound statement s contains n component statements, there will need to be 2 n rows in the truth table for s.

7. Steps to Construct a Truth Table Step1: The first n columns of the table are labeled by the component propositional variables. Further columns are included for all intermediate combinations of the variables. Step2: Under each of the first n headings, we list the 2 n possible n-tuples of truth values for the n component statements. Step3: For each of the remaining columns, we compute, in sequence, the remaining truth values.

8. Quantifiers predicate, P(x), also called a propositional function: each choice of x produces a proposition P(x) that is either true or false universal quantification of a predicate P(x): “For all values of x, P(x) is true”,  x P(x) , universal quantifier existential quantification of a predicate P(x): “There exists a value x for which P(x) is true”,  x P(x) , existential quantifier When applying both universal and existential quantification to a predicate with several variables, the order does matter

9. Conditional Statements p, q are statements, p  q is a conditional statement, or implication; p is the antecedent or hypothesis, q is the consequent or conclusion. p  q: p implies q; q, if p; p only if q; p is a sufficient condition for q; q is a necessary condition for p the converse of p  q is q  p the contrapositive of p  q is ~q  ~p p if and only if q, p  q is an equivalence or biconditional, p is a necessary and sufficient condition for q. a tautology is a statement that is true for all possible values of its propositional variables a contradiction or an absurdity is a statement that is always false a contingency is a statement that can be either true of false

10. p and q are (logically) equivalent if p  q is a tautology, denoted as p  q (p  q)  (~q  ~p) is a tautology Theorem 1.p  q  q  p; p  q  q  p; 2.p  (q  r)  (p  q)  r; p  (q  r)  (p  q)  r 3.p  (q  r)  (p  q)  (p  r); p  (q  r)  (p  q)  (p  r) 4.p  p  p; p  p  p 5.~(~p)  p; 6.De Morgan’s laws ~(p  q)  (~p)  (~q); ~(p  q)  (~p)  (~q) Conditional Statements

11. Theorem (a)(p  q)  ((~p)  q) (b)(p  q)  (~q  ~p) (c)(p  q)  ((p  q)  (q  p)) (d)~(p  q)  (p  ~q) (e)~(p  q)  ((p  ~q)  (q  ~p)) Conditional Statements

12. Theorem. Each of the following is a tautology. (a)(p  q)  p; (p  q)  q (b)p  (p  q); q  (p  q) (c)~p  (p  q); ~(p  q)  p) (d)(p  (p  q))  q; (~p  (p  q))  q (e)(~q  (p  q))  ~p (f)((p  q)  (q  r))  (p  r)

13. Theorem (a)~(  x P(x))   x(~P(x)) (b)~(  x P(x))   x(~P(x)) (c)  x(P(x)  Q(x))   x P(x)   x Q(x) (d)  x(P(x)  Q(x))   x P(x)   x Q(x) (e)  x(P(x)  Q(x))   x P(x)   x Q(x) (f)((  x P(x))  (  x Q(x)))   x (P(x)  Q(x)) is a tautology. (g)  x(P(x)  Q(x))   x P(x)   x Q(x) is a tautology. Tautologies for Quantifiers