Valid Arguments Decide if each argument is valid. If the stones are rolling, they are not gathering moss. If the stones are not gathering moss, they are a smooth group of rocks. Therefore, if the stones are rolling, they are a smooth rock group. If a penny has an Indian head on it, it is very old. If a penny has an Indian head on it, it is worth more than one cent. Therefore, if a penny is very old, it is worth more than one cent.

Valid Arguments, continued If your name is in Who’s Who, then you know what’s what. If you are not sure of where’s where, then you don’t know what’s what. If your name is in Who’s Who, then you are sure of where’s where.

 Statement - A sentence which is either true or false.  and: p and q, p  q  or: p or q, p  q  implies (conditional statement, if p, then q): p  q  if and only if (biconditional statement, p iff q), p  q, equivalent to p  q and q  p.  not p, the negation of p, ~ p Logic

Combining Statements  Conditional Statement: if p then q, p  q  Converse of a conditional statement: if q, then p, q  p  Contrapositive of conditional: if not q, then not p, ~ q  ~ p  Inverse of conditional: if not p, then not q, ~ p  ~ q

Constructing Truth Tables A truth table tells exactly under what circumstances a statement is true and a statement is false. All true-false combinations must be considered, 2 n combinations. Equivalent statements are statements that are true at the same time and false at the same time. If p is true,  p is false.

Equivalent Statements p  q is equivalent to p  q  q  p. ~ ( p  q )  (~ p )  ( ~ q ) Activity 1.7 ~ ( p  q )  (~ p )  ( ~ q ) Activity 1.8 ( p  q )  r  p  ( q  r ) (1.9) Parentheses are needed! ( p  q )  r  p  ( q  r ) (1.10) What property does this look like? p  ( q  r )  ? (1.12) p  ( q  r )  ? What property does this look like? p  ( q  r )  p  ~ q  r (1.14)

Logic of a 5th Grader = n%3fsource=search_video = n%3fsource=search_video

Open Sentences and Quantifiers An open sentence contains one or more variables and becomes a statement when the variable is replaced.  is a universal quantifier which means: for every or for all.  (x), P(x)… is read: For all x, P(x).  is an existential quantifier which means: there exists, for some, for at least one.  (x), P(x)… is read: There exists an x such that P(x).

How many teachers are there? King Middle School has 1200 students. Each student takes 6 classes in a day. Each class has 30 students and one teacher. Every teacher teaches five classes and has one planning period in a day. Each period there are 40 classes. Every student has to take a class during each of the six periods.

Rational and Irrational Numbers The need for multiplicative inverses and rational numbers The need for irrational numbers Prove √ 2 is an irrational number. (Pg. 159) Real Numbers - Filling the holes on the number line Complex Numbers - Solution for x = 0 √-1 = i, a + bi, where a,b  R

Symbols for Sets of Numbers Natural, Counting Numbers N + Whole Numbers IntegersZ Rational Numbers (a/b, b≠0, a,b  Z) Q Positive Rational NumbersQ + Real Numbers (real number line) R Positive Real NumbersR + Complex Numbers (a + bi, a,b  R) C

Proof by Contradiction You are taking a true-false quiz with 5 questions. From past experience you know: If the first answer is true, the next one is false. The last answer is always the same as the first answer. You are positive the second answer is true. On the assumption that these statements are correct, prove that the last answer is false.

Describing Sets Roster: {...-4, -2, 0, 2, 4, 6...} Set Builder Notation: {x  P(x) = 2x, x  Z} Words: {x  x is an even integer}

Sets - Symbols and Definitions A  B; A is a subset of B. A = B; A equals B { }, Ø; Empty Set A  B; {x  x  A or x  B} A  B; {x  x  A and x  B} B; Complement of B A - B; Complement of B in A, {x  x  A and x  B} A  B; Symmetric Difference, (A - B)  (B - A) A  B; Cartesian Product; {(a, b)  a  A and B  B} n(A) n(B) = n(A  B)

Theorems about Sets Draw Venn Diagrams to justify DeMorgan’s Laws A – (B  C) = (A – B)  (A – C) A – (B  C) = (A – B)  (A – C)

Using the Graphing Calculator Rule ( X KEY, Y = ) Table ( TBLSET, TABLE ) Graph ( WINDOW, TRACE, MODE, FORMAT )