Download Mathematical Logic

Mathematical Logic
Adapted from Discrete Math
Learning Objectives
• Learn about sets
• Explore various operations on sets
• Become familiar with Venn diagrams
• Learn how to represent sets in computer memory
• Learn about statements (propositions)
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Learning Objectives
• Learn how to use logical connectives to combine
statements
• Explore how to draw conclusions using various
argument forms
• Become familiar with quantifiers and predicates
• Learn various proof techniques
• Explore what an algorithm is
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Mathematical Logic
• Definition: Methods of reasoning, provides rules and
techniques to determine whether an argument is
valid
• Theorem: a statement that can be shown to be true
(under certain conditions)
– Example: If x is an even integer, then x + 1 is an odd
integer
• This statement is true under the condition that x is
an integer is true
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Mathematical Logic
• A statement, or a proposition, is a declarative
sentence that is either true or false, but not both
• Lowercase letters denote propositions
– Examples:
• p: 2 is an even number (true)
• q: 3 is an odd number (true)
• r: A is a consonant (false)
– The following are not propositions:
• p: My cat is beautiful
• q: Are you in charge?
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Mathematical Logic
• Truth value
– One of the values “truth” or “falsity” assigned to a statement
– True is abbreviated to T or 1
– False is abbreviated to F or 0
• Negation
– The negation of p, written ∼p, is the statement obtained by
negating statement p
• Truth values of p and ∼p are opposite
• Symbol ~ is called “not” ~p is read as as “not p”
• Example:
– p: A is a consonant
– ~p: it is the case that A is not a consonant
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Mathematical Logic
• Truth Table
• Conjunction
– Let p and q be statements.The conjunction of p and q,
written p ^ q , is the statement formed by joining statements p
and q using the word “and”
– The statement p∧q is true if both p and q are true;
otherwise p∧q is false
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Mathematical Logic
• Conjunction
– Truth Table for
Conjunction:
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Mathematical Logic
• Disjunction
– Let p and q be statements. The disjunction of p and q,
written p ∨ q , is the statement formed by joining
statements p and q using the word “or”
– The statement p∨q is true if at least one of the
statements p and q is true; otherwise p∨q is false
– The symbol ∨ is read “or”
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Mathematical Logic
• Disjunction
– Truth Table for
Disjunction:
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Mathematical Logic
• Implication
– Let p and q be statements.The statement “if p then q” is
called an implication or condition.
– The implication “if p then q” is written p  q
– p  q is read:
• “If p, then q”
• “p is sufficient for q”
• q if p
• q whenever p
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Mathematical Logic
• Implication
– Truth Table for Implication:
– p is called the hypothesis, q is called the
conclusion
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Mathematical Logic
• Implication
– Let p: Today is Sunday and q: I will wash the car. The
conjunction p  q is the statement:
• p  q : If today is Sunday, then I will wash the car
– The converse of this implication is written q  p
• If I wash the car, then today is Sunday
– The inverse of this implication is ~p  ~q
• If today is not Sunday, then I will not wash the car
– The contrapositive of this implication is ~q  ~p
• If I do not wash the car, then today is not Sunday
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Mathematical Logic
• Biimplication
– Let p and q be statements. The statement “p if and
only if q” is called the biimplication or biconditional
of p and q
– The biconditional “p if and only if q” is written p  q
– p  q is read:
•
•
•
•
“p if and only if q”
“p is necessary and sufficient for q”
“q if and only if p”
“q when and only when p”
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Mathematical Logic
• Biconditional
– Truth Table for the Biconditional:
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Mathematical Logic
•
Statement Formulas
–
Definitions
•
•
Symbols p ,q ,r ,...,called statement variables
Symbols ~, ∧, ∨, →,and ↔ are called logical
connectives
1) A statement variable is a statement formula
2) If A and B are statement formulas, then the
expressions (~A ), (A ∧ B) , (A ∨ B ), (A → B ) and (A
↔ B ) are statement formulas
• Expressions are statement formulas that are
constructed only by using 1) and 2) above
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Mathematical Logic
• Precedence of logical connectives is:
– ~ highest
– ∧ second highest
– ∨ third highest
– → fourth highest
– ↔ fifth highest
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Mathematical Logic
• Example:
– Let A be the statement formula (~(p ∨q )) → (q
∧p )
– Truth Table for A is:
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Mathematical Logic
• Tautology
– A statement formula A is said to be a tautology if the
truth value of A is T for any assignment of the truth
values T and F to the statement variables occurring in
A
• Contradiction
– A statement formula A is said to be a contradiction if
the truth value of A is F for any assignment of the
truth values T and F to the statement variables
occurring in A
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Mathematical Logic
• Logically Implies
– A statement formula A is said to logically imply a
statement formula B if the statement formula A → B is
a tautology. If A logically implies B, then symbolically
we write A → B
• Logically Equivalent
– A statement formula A is said to be logically
equivalent to a statement formula B if the statement
formula A ↔ B is a tautology. If A is logically
equivalent to B , then symbolically we write A ≡ B (or
A ⇔ B)
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Mathematical Logic
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• Next slide, adapted from
National Taiwan University
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