Intro to Mathematical Reasoning via Discrete Mathematics CMSC-37115 Instructor: Laszlo Babai University of Chicago Week 2, Tuesday, October 6, 2020 CMSC-37115 Mathematical Reasoning
Functions f : A → B assigns value f ( a ) ∈ B to each a ∈ A domain: set A codomain: set B CMSC-37115 Mathematical Reasoning
Functions f : A → B assigns value f ( a ) ∈ B to each a ∈ A domain: set A codomain: set B ( ∀ a ∈ A )( ∃ ! b ∈ B )( f ( a ) = b ) CMSC-37115 Mathematical Reasoning
Functions f : A → B assigns value f ( a ) ∈ B to each a ∈ A domain: set A codomain: set B range: range ( f ) = { f ( a ) | a ∈ A } values actually taken CMSC-37115 Mathematical Reasoning
Functions f : A → B assigns value f ( a ) ∈ B to each a ∈ A domain: set A codomain: set B range ( f ) = { f ( a ) | a ∈ A } range: values actually taken range ( f ) ⊆ B CMSC-37115 Mathematical Reasoning
Functions f : A → B assigns value f ( a ) ∈ B to each a ∈ A domain: set A codomain: set B range: range ( f ) = { f ( a ) | a ∈ A } values actually taken Example: A = { Alabama, Alaska, Arizona, Arkansas, California, . . . , Wisconsin, Wyoming } B = { 3 , 4 , . . . , 538 } Table: el ( x ) : number of electors from state x x AL AK AZ AR CA CO CT DE DC FL GA . . . WI WY el ( x ) 9 3 11 6 55 9 7 3 3 29 16 10 3 range ( el ) = { 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 18 , 20 , 29 , 38 , 55 } CMSC-37115 Mathematical Reasoning
Functions f : A → B assigns value f ( a ) ∈ B to each a ∈ A domain: set A codomain: set B range: range ( f ) = { f ( a ) | a ∈ A } values actually taken Example: A = { Alabama, Alaska, Arizona, Arkansas, California, . . . , Wisconsin, Wyoming } B = { 3 , 4 , . . . , 538 } Table: el ( x ) : number of electors from state x x AL AK AZ AR CA CO CT DE DC FL GA . . . WI WY el ( x ) 9 3 11 6 55 9 7 3 3 29 16 10 3 range ( el ) = { 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 18 , 20 , 29 , 38 , 55 } | range ( el ) | = 19 CMSC-37115 Mathematical Reasoning
Injection, surjection, bijection Notation: B A set of A → B functions f ∈ B A injective if there are no collisions, i.e., ( ∀ u , v ∈ A )( f ( u ) = f ( v ) ⇒ u = v ) f ∈ B A surjective if range ( f ) = B , i.e., ( ∀ z ∈ B )( ∃ u ∈ A )( f ( u ) = z ) CMSC-37115 Mathematical Reasoning
Injection, surjection, bijection Notation: B A set of A → B functions f ∈ B A injective if there are no collisions, i.e., ( ∀ u , v ∈ A )( f ( u ) = f ( v ) ⇒ u = v ) f ∈ B A surjective if range ( f ) = B , i.e., ( ∀ z ∈ B )( ∃ u ∈ A )( f ( u ) = z ) f ∈ B A bijective if injective and surjective, i.e., ( ∀ z ∈ B )( ∃ ! u ∈ A )( f ( u ) = z ) CMSC-37115 Mathematical Reasoning
Injection, surjection, bijection Notation: B A set of A → B functions f ∈ B A injective if there are no collisions, i.e., ( ∀ u , v ∈ A )( f ( u ) = f ( v ) ⇒ u = v ) f ∈ B A surjective if range ( f ) = B , i.e., ( ∀ z ∈ B )( ∃ u ∈ A )( f ( u ) = z ) f ∈ B A bijective if injective and surjective, i.e., ( ∀ z ∈ B )( ∃ ! u ∈ A )( f ( u ) = z ) existence of u means CMSC-37115 Mathematical Reasoning
Injection, surjection, bijection Notation: B A set of A → B functions f ∈ B A injective if there are no collisions, i.e., ( ∀ u , v ∈ A )( f ( u ) = f ( v ) ⇒ u = v ) f ∈ B A surjective if range ( f ) = B , i.e., ( ∀ z ∈ B )( ∃ u ∈ A )( f ( u ) = z ) f ∈ B A bijective if injective and surjective, i.e., ( ∀ z ∈ B )( ∃ ! u ∈ A )( f ( u ) = z ) existence of u means surjectivity CMSC-37115 Mathematical Reasoning
Injection, surjection, bijection Notation: B A set of A → B functions f ∈ B A injective if there are no collisions, i.e., ( ∀ u , v ∈ A )( f ( u ) = f ( v ) ⇒ u = v ) f ∈ B A surjective if range ( f ) = B , i.e., ( ∀ z ∈ B )( ∃ u ∈ A )( f ( u ) = z ) f ∈ B A bijective if injective and surjective, i.e., ( ∀ z ∈ B )( ∃ ! u ∈ A )( f ( u ) = z ) existence of u means surjectivity uniqueness of u means injectivity non-uniqueness means collision: f ( u 1 ) = f ( u 2 ) = z CMSC-37115 Mathematical Reasoning
Injection, surjection, bijection Notation: f : A → B (domain → codomain) if f ( a ) = b then write f : a �→ b ( a maps to b ) L A T EX a \ mapsto b CMSC-37115 Mathematical Reasoning
Injection, surjection, bijection Notation: f : A → B (domain → codomain) if f ( a ) = b then write f : a �→ b ( a maps to b ) L A T EX a \ mapsto b √ f ( x ) = x 2 Example: − 3 �→ 9, 2 �→ 2 CMSC-37115 Mathematical Reasoning
Injection, surjection, bijection Notation: f : A → B (domain → codomain) if f ( a ) = b then write f : a �→ b ( a maps to b ) L A T EX a \ mapsto b √ f ( x ) = x 2 Example: − 3 �→ 9, 2 �→ 2 Terminology function = map or mapping injective function = injective map = injection surjective function = surjective map = surjection bijective function = bijective map = bijection CMSC-37115 Mathematical Reasoning
Comparing cardinalities If ∃ A → B surjection then | A | ≥ | B | If ∃ A → B injection then | A | ≤ | B | CMSC-37115 Mathematical Reasoning
Comparing cardinalities If ∃ A → B surjection then | A | ≥ | B | If ∃ A → B injection then | A | ≤ | B | If ∃ A → B bijection then | A | = | B | CMSC-37115 Mathematical Reasoning
Comparing cardinalities If ∃ A → B surjection then | A | ≥ | B | If ∃ A → B injection then | A | ≤ | B | If ∃ A → B bijection then | A | = | B | Example. A , B ⊆ Z Sumset A + B = { a + b | a ∈ A , b ∈ B } CMSC-37115 Mathematical Reasoning
Comparing cardinalities If ∃ A → B surjection then | A | ≥ | B | If ∃ A → B injection then | A | ≤ | B | If ∃ A → B bijection then | A | = | B | Example. A , B ⊆ Z Sumset A + B = { a + b | a ∈ A , b ∈ B } Proposition (little theorem) | A + B | ≤ | A | · | B | CMSC-37115 Mathematical Reasoning
Comparing cardinalities If ∃ A → B surjection then | A | ≥ | B | If ∃ A → B injection then | A | ≤ | B | If ∃ A → B bijection then | A | = | B | Example. A , B ⊆ Z Sumset A + B = { a + b | a ∈ A , b ∈ B } Proposition (little theorem) | A + B | ≤ | A | · | B | Proof via surjection We know: | A × B | = | A | · | B | . Goal: Find surjection f : A × B → A + B . CMSC-37115 Mathematical Reasoning
Comparing cardinalities If ∃ A → B surjection then | A | ≥ | B | If ∃ A → B injection then | A | ≤ | B | If ∃ A → B bijection then | A | = | B | Example. A , B ⊆ Z Sumset A + B = { a + b | a ∈ A , b ∈ B } Proposition (little theorem) | A + B | ≤ | A | · | B | Proof via surjection We know: | A × B | = | A | · | B | . Goal: Find surjection f : A × B → A + B . So for ( a , b ) ∈ A × B , need to define f ( a , b ) CMSC-37115 Mathematical Reasoning
Comparing cardinalities If ∃ A → B surjection then | A | ≥ | B | If ∃ A → B injection then | A | ≤ | B | If ∃ A → B bijection then | A | = | B | Example. A , B ⊆ Z Sumset A + B = { a + b | a ∈ A , b ∈ B } Proposition (little theorem) | A + B | ≤ | A | · | B | Proof via surjection We know: | A × B | = | A | · | B | . Goal: Find surjection f : A × B → A + B . So for ( a , b ) ∈ A × B , need to define f ( a , b ) CHAT! CMSC-37115 Mathematical Reasoning
Comparing cardinalities If ∃ A → B surjection then | A | ≥ | B | If ∃ A → B injection then | A | ≤ | B | If ∃ A → B bijection then | A | = | B | Example. A , B ⊆ Z Sumset A + B = { a + b | a ∈ A , b ∈ B } Proposition (little theorem) | A + B | ≤ | A | · | B | Proof via surjection We know: | A × B | = | A | · | B | . Goal: Find surjection f : A × B → A + B . So for ( a , b ) ∈ A × B , need to define f ( a , b ) CHAT! f ( a , b ) := a + b CMSC-37115 Mathematical Reasoning
Comparing cardinalities If ∃ A → B surjection then | A | ≥ | B | If ∃ A → B injection then | A | ≤ | B | If ∃ A → B bijection then | A | = | B | Example. A , B ⊆ Z Sumset A + B = { a + b | a ∈ A , b ∈ B } Proposition (little theorem) | A + B | ≤ | A | · | B | Proof via surjection We know: | A × B | = | A | · | B | . Goal: Find surjection f : A × B → A + B . So for ( a , b ) ∈ A × B , need to define f ( a , b ) CHAT! f ( a , b ) := a + b Why is this function surjective? The definition of A + B says: A + B = range ( f ) . (Check!) CMSC-37115 Mathematical Reasoning
Comparison via injection If ∃ A → B injection then | A | ≤ | B | CMSC-37115 Mathematical Reasoning
Comparison via injection If ∃ A → B injection then | A | ≤ | B | This is a famous “principle.” What is it called? CMSC-37115 Mathematical Reasoning
Comparison via injection If ∃ A → B injection then | A | ≤ | B | This is a famous “principle.” What is it called? Pigeon Hole Principle. If | A | > | B | then every function f : A → B has a collision. CMSC-37115 Mathematical Reasoning
Comparison via injection If ∃ A → B injection then | A | ≤ | B | This is a famous “principle.” What is it called? Pigeon Hole Principle. If | A | > | B | then every function f : A → B has a collision. Notation: For n ≥ 0 we write [ n ] = { 1 , . . . , n } Examples: [ 3 ] = { 1 , 2 , 3 } , [ 1 ] = { 1 } , [ 0 ] = CMSC-37115 Mathematical Reasoning
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