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**Adapted from Discrete Math**

Mathematical Logic Adapted from Discrete Math

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**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) dww-logic

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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 dww-logic

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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 dww-logic

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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? dww-logic

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**Mathematical Logic Truth value Negation**

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 dww-logic

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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 dww-logic

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Mathematical Logic Conjunction Truth Table for Conjunction: dww-logic

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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” dww-logic

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Mathematical Logic Disjunction Truth Table for Disjunction: dww-logic

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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 dww-logic

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**Mathematical Logic Implication Truth Table for Implication:**

p is called the hypothesis, q is called the conclusion dww-logic

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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 dww-logic

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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” dww-logic

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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 A statement variable is a statement formula 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 dww-logic

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**Mathematical Logic Precedence of logical connectives is:**

~ highest ∧ second highest ∨ third highest → fourth highest ↔ fifth highest dww-logic

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**Mathematical Logic Example:**

Let A be the statement formula (~(p ∨q )) → (q ∧p ) Truth Table for A is: dww-logic

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**Mathematical Logic Tautology Contradiction**

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 dww-logic

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**Mathematical Logic Logically Implies Logically Equivalent**

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) dww-logic

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Mathematical Logic dww-logic

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**Next slide, adapted from National Taiwan University**

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