Intro to Mathematical Reasoning via Discrete Mathematics CMSC-37115 Instructor: Laszlo Babai University of Chicago Week 1, Tuesday, September 29, 2020 CMSC-37115 Mathematical Reasoning
Sets, functions, numbers Sets of numbers: N = { 1 , 2 , 3 , . . . } natural numbers Z = { . . . , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , . . . } integers Q rational numbers R real numbers C complex numbers CMSC-37115 Mathematical Reasoning
Some facts from basic arithmetic Two operations: addition, multiplication Identities a + b = b + a commutativity of addition ab = ba commutativity of multiplication ( a + b ) + c = a + ( b + c ) associativity of addition ( ab ) c = a ( bc ) associativity of multiplication An identity that connects the two operations? CMSC-37115 Mathematical Reasoning
Some facts from basic arithmetic Two operations: addition, multiplication Identities a + b = b + a commutativity of addition ab = ba commutativity of multiplication ( a + b ) + c = a + ( b + c ) associativity of addition ( ab ) c = a ( bc ) associativity of multiplication An identity that connects the two operations? a ( b + c ) = ab + ac distributivity CMSC-37115 Mathematical Reasoning
Sets, functions, numbers Ordering (for Z , R ): a ≥ b CMSC-37115 Mathematical Reasoning
Sets, functions, numbers Ordering (for Z , R ): a ≥ b Most important property of ordering: If a ≥ b and b ≥ c then a ≥ c CMSC-37115 Mathematical Reasoning
Sets, functions, numbers Ordering (for Z , R ): a ≥ b Most important property of ordering: If a ≥ b and b ≥ c then a ≥ c Notation ( a ≥ b ) ∧ ( b ≥ c ) ⇒ ( a ≥ c ) What is this property called? CMSC-37115 Mathematical Reasoning
Sets, functions, numbers Ordering (for Z , R ): a ≥ b Most important property of ordering: If a ≥ b and b ≥ c then a ≥ c Notation ( a ≥ b ) ∧ ( b ≥ c ) ⇒ ( a ≥ c ) What is this property called? transitivity of the “ ≥ ” relation CMSC-37115 Mathematical Reasoning
Sets, functions, numbers Ordering (for Z , R ): a ≥ b Most important property of ordering: If a ≥ b and b ≥ c then a ≥ c Notation ( a ≥ b ) ∧ ( b ≥ c ) ⇒ ( a ≥ c ) What is this property called? transitivity of the “ ≥ ” relation Relation between ordering and addition? CMSC-37115 Mathematical Reasoning
Sets, functions, numbers Ordering (for Z , R ): a ≥ b Most important property of ordering: If a ≥ b and b ≥ c then a ≥ c Notation ( a ≥ b ) ∧ ( b ≥ c ) ⇒ ( a ≥ c ) What is this property called? transitivity of the “ ≥ ” relation Relation between ordering and addition? ( a ≥ b ) ⇒ ( a + c ≥ b + c ) CMSC-37115 Mathematical Reasoning
Number theory Focus on Z (for a while) Central concept: divisibility “8 divides 48” and “8 does not divide 28” 8 ∤ 28 Notation 8 | 48, Terminology. All of the following statements are synonymous: 8 divides 48 8 is a divisor of 48 48 is a multiple of 8 48 is divisible by 8 Their common abbreviation is 8 | 48. More examples: 9 | 63, 9 | − 63, 7 | 63, 37 | 999 CMSC-37115 Mathematical Reasoning
Divisibility Definition When do we say that a | b ? CMSC-37115 Mathematical Reasoning
Divisibility Definition When do we say that a | b ? a | b if there exists x such that ax = b Notation ( ∃ x )( ax = b ) We call such an x a witness (to the validity of the satement ( ∃ x )( ax = b ) ∃ – existential quantifier “there exists . . . such that” CMSC-37115 Mathematical Reasoning
Divisibility Definition When do we say that a | b ? a | b if there exists x such that ax = b Notation ( ∃ x )( ax = b ) We call such an x a witness (to the validity of the satement ( ∃ x )( ax = b ) ∃ – existential quantifier “there exists . . . such that” ∀ – universal quantifier “for all” ( ∀ a , b )( a + b = b + a ) CMSC-37115 Mathematical Reasoning
Quantifiers ∃ – existential quantifier “there exists . . . such that” ∀ – universal quantifier “for all” CMSC-37115 Mathematical Reasoning
Quantifiers ∃ – existential quantifier “there exists . . . such that” ∀ – universal quantifier “for all” True or false? ( ∀ a )( ∃ x )( ax = 1 ) CMSC-37115 Mathematical Reasoning
Quantifiers ∃ – existential quantifier “there exists . . . such that” ∀ – universal quantifier “for all” True or false? ( ∀ a )( ∃ x )( ax = 1 ) False: a = 0 is a counterexample: ( ∄ x )( 0 · x = 1 ) CMSC-37115 Mathematical Reasoning
Quantifiers ∃ – existential quantifier “there exists . . . such that” ∀ – universal quantifier “for all” True or false? ( ∀ a )( ∃ x )( ax = 1 ) False: a = 0 is a counterexample: ( ∄ x )( 0 · x = 1 ) Tue or false? ( ∀ a )( a � 0 ⇒ ( ∃ x )( ax = 1 )) CMSC-37115 Mathematical Reasoning
Quantifiers ∃ – existential quantifier “there exists . . . such that” ∀ – universal quantifier “for all” True or false? ( ∀ a )( ∃ x )( ax = 1 ) False: a = 0 is a counterexample: ( ∄ x )( 0 · x = 1 ) Tue or false? ( ∀ a )( a � 0 ⇒ ( ∃ x )( ax = 1 )) Depends of the universe over which the quantifiers range CMSC-37115 Mathematical Reasoning
Quantifiers ∃ – existential quantifier “there exists . . . such that” ∀ – universal quantifier “for all” True or false? ( ∀ a )( ∃ x )( ax = 1 ) False: a = 0 is a counterexample: ( ∄ x )( 0 · x = 1 ) Tue or false? ( ∀ a )( a � 0 ⇒ ( ∃ x )( ax = 1 )) Depends of the universe over which the quantifiers range True if the universe is R , false if the universe is Z CMSC-37115 Mathematical Reasoning
Quantifiers ∃ – existential quantifier “there exists . . . such that” ∀ – universal quantifier “for all” True or false? ( ∀ a )( ∃ x )( ax = 1 ) False: a = 0 is a counterexample: ( ∄ x )( 0 · x = 1 ) Tue or false? ( ∀ a )( a � 0 ⇒ ( ∃ x )( ax = 1 )) Depends of the universe over which the quantifiers range True if the universe is R , false if the universe is Z ( ∀ a ∈ R )(( a � 0 ) ⇒ ( ∃ x ∈ R )( ax = 1 )) TRUE ( ∀ a ∈ Z )(( a � 0 ) ⇒ ( ∃ x ∈ Z )( ax = 1 )) FALSE CMSC-37115 Mathematical Reasoning
Quantifiers ∃ – existential quantifier “there exists . . . such that” ∀ – universal quantifier “for all” True or false? ( ∀ a )( ∃ x )( ax = 1 ) False: a = 0 is a counterexample: ( ∄ x )( 0 · x = 1 ) Tue or false? ( ∀ a )( a � 0 ⇒ ( ∃ x )( ax = 1 )) Depends of the universe over which the quantifiers range True if the universe is R , false if the universe is Z ( ∀ a ∈ R )(( a � 0 ) ⇒ ( ∃ x ∈ R )( ax = 1 )) TRUE ( ∀ a ∈ Z )(( a � 0 ) ⇒ ( ∃ x ∈ Z )( ax = 1 )) FALSE Mixed universes ( ∀ a ∈ Z )(( a � 0 ) ⇒ ( ∃ x ∈ R )( ax = 1 )) TRUE CMSC-37115 Mathematical Reasoning
Proof Let us fix Z as our universe. How do we prove that this statement is false: ( ∀ a )( a � 0 ⇒ ( ∃ x )( ax = 1 )) We need to find a counterexample, i.e., need to show ( ∃ a )( a � 0 � ( ∃ x )( ax = 1 )) What would be such an a ? CMSC-37115 Mathematical Reasoning
Proof Let us fix Z as our universe. How do we prove that this statement is false: ( ∀ a )( a � 0 ⇒ ( ∃ x )( ax = 1 )) We need to find a counterexample, i.e., need to show ( ∃ a )( a � 0 � ( ∃ x )( ax = 1 )) What would be such an a ? a = 2, for instance. ( 2 � 0 ) � ( ∃ x )( ax = 1 )) When is an inference A ⇒ B false? If A is true but B is false CMSC-37115 Mathematical Reasoning
Proof Let us fix Z as our universe. How do we prove that this statement is false: ( ∀ a )( a � 0 ⇒ ( ∃ x )( ax = 1 )) We need to find a counterexample, i.e., need to show ( ∃ a )( a � 0 � ( ∃ x )( ax = 1 )) What would be such an a ? a = 2, for instance. ( 2 � 0 ) � ( ∃ x )( ax = 1 )) When is an inference A ⇒ B false? If A is true but B is false In our case, A is true ( 2 � 0 ) ; we need to show that B is false, i.e., ( ∄ x )( 2 x = 1 ) , i.e., ( ∀ x )( 2 x � 1 ) . Which is obvious: if x ≤ 0 then 2 x ≤ 0, and if x ≥ 1 then 2 x ≥ 2. � CMSC-37115 Mathematical Reasoning
Quantifiers Negation of a universally quantified formula begins with an existential quantifier and vice versa CMSC-37115 Mathematical Reasoning
Divisibility “ a divides b ” a | b e.g., 8 | 24 Recall DEF: a | b if ( ∃ x )( ax = b ) A little theorem: ( ∀ a , b , c )(( a | b ) ∧ ( b | c ) ⇒ ( a | c )) the “divisibility” relation is CMSC-37115 Mathematical Reasoning
Divisibility “ a divides b ” a | b e.g., 8 | 24 Recall DEF: a | b if ( ∃ x )( ax = b ) A little theorem: ( ∀ a , b , c )(( a | b ) ∧ ( b | c ) ⇒ ( a | c )) the “divisibility” relation is transitive CMSC-37115 Mathematical Reasoning
Divisibility “ a divides b ” a | b e.g., 8 | 24 Recall DEF: a | b if ( ∃ x )( ax = b ) A little theorem: ( ∀ a , b , c )(( a | b ) ∧ ( b | c ) ⇒ ( a | c )) the “divisibility” relation is transitive Proof. Assumptions: ( ∃ x )( ax = b ) and ( ∃ y )( by = c ) What is the desired conclusion? CMSC-37115 Mathematical Reasoning
Divisibility “ a divides b ” a | b e.g., 8 | 24 Recall DEF: a | b if ( ∃ x )( ax = b ) A little theorem: ( ∀ a , b , c )(( a | b ) ∧ ( b | c ) ⇒ ( a | c )) the “divisibility” relation is transitive Proof. Assumptions: ( ∃ x )( ax = b ) and ( ∃ y )( by = c ) What is the desired conclusion? ( ∃ z )( az = c ) CMSC-37115 Mathematical Reasoning
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