1.1 Sets and Logic Set – a collection of objects. Set brackets {} are used to enclose the elements of a set. Example: {1, 2, 5, 9} Elements – objects inside the brackets 2  A means 2 is an element of set A 3  A means 3 is not an element of set A Cardinal number – number of elements of a set notation: n(A) = # elements in set A

1.1 Sets and Logic Sets are equal – they contain the same elements (the order can be different) example: {A, B, C} = {B, C, A} {x | x has the property y} – This is read: “The set of x such that x has the property y” examples: {x | x is a letter grade} {x | x is an integer between –1.5 and 5.2}

1.1 Sets and Logic Universal set – set of all elements in a given situation example: all outcomes when a die is rolled U = {1, 2, 3, 4, 5, 6} Empty set – set of no elements, denoted by  Subset – B  A (B is a subset of A) true if every element of B is also an element of A Proper subset – B  A (B is a proper subset of A) true if B  A and B  A

1.1 Sets and Logic For all sets:   A and A  A # of subsets – a set with n distinct elements has 2 n subsets {  } is different from  ;  = {}  has no elements (cardinality = 0) {  } has one element (cardinality = 1)

1.1 Sets and Logic Pascal’s triangle can be used to find the number of subsets with a given number of elements.

1.2 Set Operations Complement of a set A – the set of all elements that are in the universal set associated with set A but not in A itself. In text: A C = complement of A example: U = {1, 2, 3, 4, 5, 6} A= {1, 2} then A C = {3, 4, 5, 6} Cardinalities: n(A) + n(A C ) = n(U) example: n(U) = 12 and n(A) = 3; find n(A C ) n(A C ) = n(U) – n(A) = 12 – 3 = 9

1.2 Set Operations Venn diagrams – useful for visualizing sets A A B ACAC A set and its complement B  A

1.2 Set Operations General Venn diagram for 2 sets A II B I If A  B region II is empty If B  A region IV is empty III IV

1.2 Set Operations Union – The union of two sets A & B is the set that contains all the elements that are in A or B or both A and B – denoted A  B (regions II, III, and IV above) Intersection – The set of all elements that are in both A and B – denoted by A  B (region III above) Disjoint sets – If 2 sets have no elements in common they are disjoint - A  B =  (region III is empty)

1.3 Sets and Venn Diagrams De Morgan’s Laws for sets: –A C  B C = (A  B) C –A C  B C = (A  B) C

1. 3 Sets and Venn Diagrams General Venn diagram for 3 sets A C B Divided into 8 regions

1.3 Sets and Venn Diagrams Venn diagram - shading A B A  B: crisscross area A  B: all shaded area

1.3 Sets and Venn Diagrams Venn diagram – disjoint sets A B

1.3 Sets and Venn Diagrams Cardinality rule for the union of 2 sets: n (A  B) = n(A) + n(B) - n(A  B ) Cardinality rule for the union of 3 sets: n(A  B  C) = n(A) + n(B) + n(C) - n(A  B ) - n(B  C ) - n(A  C ) + n(A  B  C)

1.4 Inductive and Deductive Logic Inductive Logic – is the process of drawing a general conclusion from specific case. Example: When a number ending in 5 is squared, does the result end in 25? 5 2 = = = = = = Inductive logic says this is true

1.4 Inductive and Deductive Logic Inductive logic sometimes gives you a false conclusion. Example: Does the expression n 2 – n + 11 always give a prime number? For n=2, n 2 – n + 11 = 13  prime For n=3, n 2 – n + 11 = 17  prime For n=4, n 2 – n + 11 = 23  prime For n=5, n 2 – n + 11 = 31  prime For n=6, n 2 – n + 11 = 41  prime

1.4 Inductive and Deductive Logic Example: Does the expression n 2 – n + 11 always give a prime number? For n=7, n 2 – n + 11 = 53  prime For n=8, n 2 – n + 11 = 67  prime For n=9, n 2 – n + 11 = 83  prime For n=10, n 2 – n + 11 = 101  prime For n=11, n 2 – n + 11 = 121 = 11 2  not prime Finally we get a counterexample!

1.4 Inductive and Deductive Logic Counterexample – a single case or example that is used to refute a mathematical conjecture Deduction – the process of drawing a specific conclusion from a general situation. Basic Syllogism (deductive logic) –2 statements (premises and a conclusion

1.4 Inductive and Deductive Logic Inductive Logic (sometimes valid) Specific cases  general case Deductive logic (always valid) General case  specific cases

1.5 Logic Statements Statement – sentence that has a truth value. The statement is either true or false but not both Negation of a statement – a statement whose truth value is always the opposite that of the original statement. The negation of P is ~P. Quantifier – a word or phrase describing the inclusiveness of the statement. Examples: some, all most, few

1.5 Logic Statements The Accord is manufactured by Honda (statement) Mathematics is the best subject (not a statement - opinion) Earth is the only planet in the universe (statement) What are fireflies? (not a statement – question) 2 – x = 3 (not a statement – equation with a variable) 1 = 2 (statement)

1.5 Logic Statements Quantifier for statement Negation AllAt least one is not SomeNone At least one is

1.5 Logic Statements Paradox – a statement or group of statements that results in a contradiction Example: “This statement is false” - it cannot be given a truth value Zeno’s Paradox – Achilles and the tortoise (on page 34 of text)

1.6 Compound Statements Definition: A truth table for a statement is a table that provides the truth value of the statement for all possible situations Definition: Two statements are logically equivalent if they have the same truth tables Definition: Conjunction of two statements p and q is the statement “p and q” – which is only true if both p and q are true. Notation: p  q

1.6 Compound Statements Definition: Disjunction of two statements p and q is the statement “p or q” – which is true if either p or q are true. Notation: p  q Truth Tables: pq p  qp  q TTTT TFTF FTTF FFFF

1.6 Compound Statements De Morgan’s Laws for negation: –~(p  q) = (~p)  (~q) –~(p  q) = (~p)  (~q)

1.7 Conditional Statements Conditional statement - can be put in the form “if p then q” (Notation: p  q) P is the antecedent or hypothesis; Q is the consequent or conclusion Truth table: pqp  q TTT TFF FTT FFT

1.7 Conditional Statements Ways to translate p  q: –If p then q –P only if q –P implies q –P is sufficient for q –Q is necessary for p –Q if p –All p are q

1.7 Conditional Statements Tautology - A compound statement that is true under all possible truth assignments. example: p  ~p Contingency - A compound statement that is sometimes true and sometimes false depending on truth assignments example: p  q Contradiction - A compound statement” that is false under all possible truth assignments example: p  ~p

1.8 More Conditionals Converse of a conditional statement - formed by interchanging the hypothesis and the conclusion. example: converse of p  q is q  p Inverse of a conditional statement - formed by negating the hypothesis and the conclusion. example: inverse of p  q is ~p  ~q Contrapositive of a conditional statement - formed by interchanging and negating the hypothesis and conclusion. example: contrapositive of p  q is ~q  ~p

1.8 More Conditionals Conditional: p  q Converse: q  p Contrapositive: ~q  ~p Inverse: ~p  ~q Rule: Interchanging and negating the hypothesis and conclusion gives an equivalent conditional

1.8 More Conditionals Biconditional statement - can be put in the form “p if and only if q” (Notation: p  q) Truth table: pqp  q TTT TFF FTF FFT

1.9 Analyzing Logical Arguments Definition: If p  q is a tautology, then q “logically follows” from p Definition: conditional representation of an argument is [p 1  p 2  p 3 ……..  p n ]  q

1.9 Analyzing Logical Arguments Direct ProofProof by contradiction Transitive Proof 1. p  q 2. p2. ~q 2. q  r qq  ~p  p  r

1.9 Analyzing Logical Arguments Definition: A fallacy is an argument that may seem to be a valid logical argument, but in fact is invalid. a = b ab = b 2 ab – a 2 = b 2 – a 2 a(b – a) = (b – a)(b + a) a = b + a a = 2a 1 = 2

1.9 Analyzing Logical Arguments Fallacy: Affirming the consequent Fallacy: Denying the antecedent 1. p  q 2. q2. ~p pp  ~q

1.9 Analyzing Logical Arguments Proof – affirming the consequent is not valid Truth table for [(p  q)  q]  p: pq p  q [(p  q)  q] [(p  q)  q]  p TTTTT TFFFT FTTTF FFTFT

1.10 Logical Circuits Definition: Switch is an electronic component that can either have power flowing through it or not. Note: This is comparable to a logic statement –Switch – “on” or “off” –Statement – “T” or “F” Definition: A group of switches connected together is a circuit

1.10 Logical Circuits Definition: “series circuit” – connection of two or more switches so that the circuit works only if both switches are on. pq

1.10 Logical Circuits Definition: “parallel circuit” – connection of two or more switches so that the circuit works if either of the switches is on. p q

1.10 Logical Circuits Definition: “complementary switches” – switches that are set up so that when one is on, the other is off and vice versa. ~p

1.10 Logical Circuits Open and closed switches: open = false, closed = true (current flows) p is open (false), q is closed (true) p q