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 ∧ ∨

connectives1) A statement variable is a statement formula2) 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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