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1.1 Sets and Logic

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

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1.1 Sets and Logic

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  1. 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 brackets2  A means 2 is an element of set A3  A means 3 is not an element of set A • Cardinal number – number of elements of a setnotation: n(A) = # elements in set A

  2. 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}

  3. 1.1 Sets and Logic • Universal set – set of all elements in a given situationexample: all outcomes when a die is rolledU = {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

  4. 1.1 Sets and Logic • For all sets:  A and A  A • # of subsets – a set with n distinct elements has 2n subsets • {} is different from ;  = {}  has no elements (cardinality = 0){} has one element (cardinality = 1)

  5. 1.1 Sets and Logic • Pascal’s triangle can be used to find the number of subsets with a given number of elements.

  6. 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: AC = complement of Aexample: U = {1, 2, 3, 4, 5, 6} A= {1, 2}then AC = {3, 4, 5, 6} • Cardinalities: n(A) + n(AC) = n(U)example: n(U) = 12 and n(A) = 3; find n(AC)n(AC) = n(U) – n(A) = 12 – 3 = 9

  7. 1.2 Set Operations • Venn diagrams – useful for visualizing sets A set and its complement AC A B  A A B

  8. 1.2 Set Operations • General Venn diagram for 2 sets If A  B region II is emptyIf B  A region IV is empty A B III II IV I

  9. 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)

  10. 1.3 Sets and Venn Diagrams • De Morgan’s Laws for sets: • ACBC = (AB)C • ACBC = (AB)C

  11. 1. 3 Sets and Venn Diagrams • General Venn diagram for 3 sets A B Divided into 8 regions C

  12. 1.3 Sets and Venn Diagrams • Venn diagram - shading A B: crisscross areaA  B: all shaded area A B

  13. 1.3 Sets and Venn Diagrams • Venn diagram – disjoint sets B A

  14. 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)

  15. 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?52 = 25152 = 225252 = 625552 = 3025952 = 90251252 = 15625Inductive logic says this is true

  16. 1.4 Inductive and Deductive Logic • Inductive logic sometimes gives you a false conclusion.Example: Does the expression n2 – n + 11 always give a prime number?For n=2, n2 – n + 11 = 13 primeFor n=3, n2 – n + 11 = 17 primeFor n=4, n2 – n + 11 = 23 primeFor n=5, n2 – n + 11 = 31 primeFor n=6, n2 – n + 11 = 41 prime

  17. 1.4 Inductive and Deductive Logic • Example: Does the expression n2 – n + 11 always give a prime number?For n=7, n2 – n + 11 = 53 primeFor n=8, n2 – n + 11 = 67 primeFor n=9, n2 – n + 11 = 83 primeFor n=10, n2 – n + 11 = 101 primeFor n=11, n2 – n + 11 = 121 = 112 not primeFinally we get a counterexample!

  18. 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

  19. 1.4 Inductive and Deductive Logic • Inductive Logic (sometimes valid)Specific cases  general case • Deductive logic (always valid)General case  specific cases

  20. 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

  21. 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)

  22. 1.5 Logic Statements

  23. 1.5 Logic Statements • Paradox – a statement or group of statements that results in a contradictionExample: “This statement is false”- it cannot be given a truth value • Zeno’s Paradox – Achilles and the tortoise (on page 34 of text)

  24. 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

  25. 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:

  26. 1.6 Compound Statements • De Morgan’s Laws for negation: • ~(p  q) = (~p)  (~q) • ~(p  q) = (~p)  (~q)

  27. 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:

  28. 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

  29. 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 assignmentsexample: pq • Contradiction - A compound statement” that is false under all possible truth assignmentsexample: p  ~p

  30. 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

  31. 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

  32. 1.8 More Conditionals • Biconditional statement - can be put in the form “p if and only if q” (Notation: pq) • Truth table:

  33. 1.9 Analyzing Logical Arguments • Definition: If pq is a tautology, then q “logically follows” from p • Definition: conditional representation of an argumentis [p1 p2  p3…….. pn]q

  34. 1.9 Analyzing Logical Arguments

  35. 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 = bab = b2 ab – a2 = b2 –a2a(b – a) = (b – a)(b + a) a = b + aa = 2a1 = 2

  36. 1.9 Analyzing Logical Arguments

  37. 1.9 Analyzing Logical Arguments • Proof – affirming the consequent is not valid • Truth table for [(p  q)  q]  p:

  38. 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

  39. 1.10 Logical Circuits • Definition: “series circuit” – connection of two or more switches so that the circuit works only if both switches are on. p q

  40. 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

  41. 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

  42. 1.10 Logical Circuits • Open and closed switches:open = false, closed = true (current flows)p is open (false), q is closed (true) p q

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