1.1 sets and logic set – a collection of objects. set brackets {} are used to enclose the elements...
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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
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 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
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)
1.1 Sets and Logic
• Pascal’s triangle can be used to find the number of subsets with a given number of elements.
193684126126843691
18285670562881
172135352171
1615201561
15101051
14641
1331
121
11
1
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
1.2 Set Operations
• Venn diagrams – useful for visualizing sets
A
A B
AC 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 emptyIII
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 AB(regions II, III, and IV above)
• Intersection – The set of all elements that are in both A and B – denoted by AB(region III above)
• Disjoint sets – If 2 sets have no elements in common they are disjoint - AB = (region III is empty)
1.3 Sets and Venn Diagrams
• De Morgan’s Laws for sets:
– ACBC = (AB)C
– ACBC = (AB)C
1. 3 Sets and Venn Diagrams
• General Venn diagram for 3 sets
A
C
BDivided into 8 regions
1.3 Sets and Venn Diagrams
• Venn diagram - shading
A BA B: crisscross area
A B: all shaded area
1.3 Sets and Venn Diagrams
• Venn diagram – disjoint sets
AB
BA
1.3 Sets and Venn Diagrams
• Cardinality rule for the union of 2 sets:n(AB) = n(A) + n(B) - n(AB)
• Cardinality rule for the union of 3 sets:n(ABC) = n(A) + n(B) + n(C) - n(AB) - n(BC) - n(AC) + n(ABC)
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
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
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 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
All At least one is not
Some None
None At least one is
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)
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:
p q p q p qT T T T
T F T F
F T T F
F F F F
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: pq)
• P is the antecedent or hypothesis; Q is the consequent or conclusion
• Truth table: p q p q
T T T
T F F
F T T
F F T
1.7 Conditional Statements
• Ways to translate pq:– 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 assignmentsexample: pq
• Contradiction - A compound statement” that is false under all possible truth assignmentsexample: p ~p
1.8 More Conditionals
• Converse of a conditional statement - formed by interchanging the hypothesis and the conclusion.example: converse of pq is qp
• Inverse of a conditional statement - formed by negating the hypothesis and the conclusion.example: inverse of pq is ~p~q
• Contrapositive of a conditional statement - formed by interchanging and negating the hypothesis and conclusion.example: contrapositive of pq is ~q~p
1.8 More Conditionals
• Conditional: pq Converse: qp
• 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: pq)
• Truth table:
p q p q
T T T
T F F
F T F
F F T
1.9 Analyzing Logical Arguments
• Definition: If pq is a tautology, then q “logically follows” from p
• Definition: conditional representation of an argumentis [p1 p2 p3…….. pn]q
1.9 Analyzing Logical Arguments
Direct Proof Proof by contradiction
Transitive Proof
1. pq 1. pq 1. pq
2. p 2. ~q 2. qr
q ~p pr
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 – a2
a(b – a) = (b – a)(b + a)a = b + a
a = 2a1 = 2
1.9 Analyzing Logical Arguments
Fallacy:Affirming the consequent
Fallacy:Denying the antecedent
1. pq 1. pq
2. q 2. ~p
p ~q
1.9 Analyzing Logical Arguments
• Proof – affirming the consequent is not valid• Truth table for [(p q) q] p:
p q p q [(p q) q] [(p q) q] p
T T T T T
T F F F T
F T T T F
F F T F T
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.
p q
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