Sets and Logic Set a collection of objects

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

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}

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

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)

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

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 A example: 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(U) – n(A) = 12 – 3 = 9

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

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

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)

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

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

Sets and Venn Diagrams • Venn diagram - shading A B A B: crisscross area A B: all shaded area

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

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)

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 = 25 152 = 225 252 = 625 552 = 3025 952 = 9025 1252 = 15625 Inductive logic says this is true

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

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 = 112 not prime Finally we get a counterexample!

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

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

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

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)

Logic Statements Quantifier for statement All Negation Some None At least one is not

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)

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

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: p q p q T T T F F F

Compound Statements • De Morgan’s Laws for negation: • ~(p q) = (~p) (~q)

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: p T q T p q T T F F T

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

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

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

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

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

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……. . pn] q

Analyzing Logical Arguments Direct Proof Transitive Proof 1. p q Proof by contradiction 1. p q 2. p 2. ~q 2. q r q ~p p r 1. p q

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 = 2 a 1=2

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

Analyzing Logical Arguments • Proof – affirming the consequent is not valid • Truth table for [(p q) q] p: p T T F F q T F p q T F T T [(p q) q] p T F T

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

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

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

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

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

Thank you!