Intro to Mathematical Reasoning via Discrete Mathematics - PowerPoint PPT Presentation

Intro to Mathematical Reasoning via Discrete Mathematics CMSC-37115 Instructor: Laszlo Babai University of Chicago Week 1, Thursday, October 1, 2020 CMSC-37115 Mathematical Reasoning

Sets, subsets setmaker bar “such that” A = { x 2 | 0 ≤ x ≤ 3 } = { 0 , 1 , 4 , 9 } CMSC-37115 Mathematical Reasoning

Sets, subsets setmaker bar “such that” A = { x 2 | 0 ≤ x ≤ 3 } = { 0 , 1 , 4 , 9 } A = { a , b , c , d , e } B = { a , b , e } = ⇒ B ⊆ A subset B ⊆ A if ( ∀ x )(( x ∈ B ) = ⇒ ( x ∈ A )) CMSC-37115 Mathematical Reasoning

Sets, subsets setmaker bar “such that” A = { x 2 | 0 ≤ x ≤ 3 } = { 0 , 1 , 4 , 9 } A = { a , b , c , d , e } B = { a , b , e } = ⇒ B ⊆ A subset B ⊆ A if ( ∀ x )(( x ∈ B ) = ⇒ ( x ∈ A )) DO The “subset” relation is CMSC-37115 Mathematical Reasoning

Sets, subsets setmaker bar “such that” A = { x 2 | 0 ≤ x ≤ 3 } = { 0 , 1 , 4 , 9 } A = { a , b , c , d , e } B = { a , b , e } = ⇒ B ⊆ A subset B ⊆ A if ( ∀ x )(( x ∈ B ) = ⇒ ( x ∈ A )) DO The “subset” relation is transitive A ⊆ B ⊆ C = ⇒ A ⊆ C CMSC-37115 Mathematical Reasoning

Sets, subsets setmaker bar “such that” A = { x 2 | 0 ≤ x ≤ 3 } = { 0 , 1 , 4 , 9 } A = { a , b , c , d , e } B = { a , b , e } = ⇒ B ⊆ A subset B ⊆ A if ( ∀ x )(( x ∈ B ) = ⇒ ( x ∈ A )) DO The “subset” relation is transitive A ⊆ B ⊆ C = ⇒ A ⊆ C A = B if A ⊆ B and B ⊆ A , i.e., A = B if ( ∀ x )( x ∈ A ⇔ x ∈ B ) Example: { a , c , a , b , c } = { a , b , c } . CMSC-37115 Mathematical Reasoning

Sets, subsets setmaker bar “such that” A = { x 2 | 0 ≤ x ≤ 3 } = { 0 , 1 , 4 , 9 } A = { a , b , c , d , e } B = { a , b , e } = ⇒ B ⊆ A subset B ⊆ A if ( ∀ x )(( x ∈ B ) = ⇒ ( x ∈ A )) DO The “subset” relation is transitive A ⊆ B ⊆ C = ⇒ A ⊆ C A = B if A ⊆ B and B ⊆ A , i.e., A = B if ( ∀ x )( x ∈ A ⇔ x ∈ B ) Example: { a , c , a , b , c } = { a , b , c } . Powerset of A : P ( A ) = { all subsets of A } CMSC-37115 Mathematical Reasoning

Operations with sets DEFINITIONS Union A ∪ B = { x | ( x ∈ A ) ∨ ( x ∈ B ) } Intersection A ∩ B = { x | ( x ∈ A ) ∧ ( x ∈ B ) } Difference A \ B = { x | ( x ∈ A ) ∧ ( x � B ) } A , B ⊆ Ω “universe” A = Ω \ A Complement CMSC-37115 Mathematical Reasoning

Operations with sets DEFINITIONS Union A ∪ B = { x | ( x ∈ A ) ∨ ( x ∈ B ) } Intersection A ∩ B = { x | ( x ∈ A ) ∧ ( x ∈ B ) } Difference A \ B = { x | ( x ∈ A ) ∧ ( x � B ) } A , B ⊆ Ω “universe” A = Ω \ A Complement CMSC-37115 Mathematical Reasoning

Indentities for set operations DO: Distributivity A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) CMSC-37115 Mathematical Reasoning

Indentities for set operations DO: Distributivity A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) De Morgan’s laws A ∪ B = A ∩ B A ∩ B = A ∪ B CMSC-37115 Mathematical Reasoning

Indentities for set operations DO: Distributivity A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ) De Morgan’s laws A ∪ B = A ∩ B A ∩ B = A ∪ B | A | cardinality of A (size of A , number of elements of A ) Example: |{ a , c , a , b , c }| = 3 CMSC-37115 Mathematical Reasoning

Indentities for cardinalities DO Modular identity | A ∩ B | + | A ∪ B | = | A | + | B | CMSC-37115 Mathematical Reasoning

Indentities for cardinalities DO Modular identity | A ∩ B | + | A ∪ B | = | A | + | B | DO | A | = n = ⇒ |P ( A ) | = CMSC-37115 Mathematical Reasoning

Indentities for cardinalities DO Modular identity | A ∩ B | + | A ∪ B | = | A | + | B | | A | = n = ⇒ |P ( A ) | = 2 n DO CMSC-37115 Mathematical Reasoning

Indentities for cardinalities DO Modular identity | A ∩ B | + | A ∪ B | = | A | + | B | | A | = n = ⇒ |P ( A ) | = 2 n DO Cartesian product A × B = { ( a , b ) | a ∈ A , b ∈ B } CMSC-37115 Mathematical Reasoning

Indentities for cardinalities DO Modular identity | A ∩ B | + | A ∪ B | = | A | + | B | | A | = n = ⇒ |P ( A ) | = 2 n DO Cartesian product A × B = { ( a , b ) | a ∈ A , b ∈ B } DO | A × B | = | A | · | B | CMSC-37115 Mathematical Reasoning

Indentities for cardinalities DO Modular identity | A ∩ B | + | A ∪ B | = | A | + | B | | A | = n = ⇒ |P ( A ) | = 2 n DO Cartesian product A × B = { ( a , b ) | a ∈ A , b ∈ B } DO | A × B | = | A | · | B | CMSC-37115 Mathematical Reasoning

Sumsets DEF shifting A ⊆ Z , b ∈ Z A + b = { a + b | a ∈ A } Example: If A = { 1 , 2 , 4 } then A + 2 = { 3 , 4 , 6 } CMSC-37115 Mathematical Reasoning

Sumsets DEF shifting A ⊆ Z , b ∈ Z A + b = { a + b | a ∈ A } Example: If A = { 1 , 2 , 4 } then A + 2 = { 3 , 4 , 6 } DEF sumset A , B ⊆ Z A + B = { a + b | a ∈ A , b ∈ B } Example B = { 2 , 4 } then A + B = ( A + 2 ) ∪ ( A + 4 ) = { 3 , 4 , 6 , 5 , 6 , 8 } = { 3 , 4 , 5 , 6 , 8 } CMSC-37115 Mathematical Reasoning

Sumsets DEF sumset A , B ⊆ Z A + B = { a + b | a ∈ A , b ∈ B } True/false | A + B | ≥ | A | CMSC-37115 Mathematical Reasoning

Sumsets DEF sumset A , B ⊆ Z A + B = { a + b | a ∈ A , b ∈ B } True/false | A + B | ≥ | A | A + ∅ = CMSC-37115 Mathematical Reasoning

Sumsets DEF sumset A , B ⊆ Z A + B = { a + b | a ∈ A , b ∈ B } True/false | A + B | ≥ | A | A + ∅ = ∅ Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≤ CMSC-37115 Mathematical Reasoning

Sumsets DEF sumset A , B ⊆ Z A + B = { a + b | a ∈ A , b ∈ B } True/false | A + B | ≥ | A | A + ∅ = ∅ Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≤ 15 CMSC-37115 Mathematical Reasoning

Sumsets DEF sumset A , B ⊆ Z A + B = { a + b | a ∈ A , b ∈ B } True/false | A + B | ≥ | A | A + ∅ = ∅ Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≤ 15 DO | A + B | ≤ | A | · | B | CMSC-37115 Mathematical Reasoning

Sumset Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≥ CMSC-37115 Mathematical Reasoning

Sumset Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≥ 7 CMSC-37115 Mathematical Reasoning

Sumset Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≥ 7 Example when | A + B | = 7 ? CMSC-37115 Mathematical Reasoning

Sumset Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≥ 7 Example when | A + B | = 7 ? A = { 0 , 1 , 2 } , B = { 0 , 1 , 2 , 3 , 4 } , A + B = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } CMSC-37115 Mathematical Reasoning

Sumset Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≥ 7 Example when | A + B | = 7 ? A = { 0 , 1 , 2 } , B = { 0 , 1 , 2 , 3 , 4 } , A + B = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } HW (1/12) | A + B | ≥ | A | + | B | − 1 assuming CMSC-37115 Mathematical Reasoning

Sumset Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≥ 7 Example when | A + B | = 7 ? A = { 0 , 1 , 2 } , B = { 0 , 1 , 2 , 3 , 4 } , A + B = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } HW (1/12) | A + B | ≥ | A | + | B | − 1 assuming A , B � ∅ 2 A := { 2 a | a ∈ A } CMSC-37115 Mathematical Reasoning

Sumset Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≥ 7 Example when | A + B | = 7 ? A = { 0 , 1 , 2 } , B = { 0 , 1 , 2 , 3 , 4 } , A + B = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } HW (1/12) | A + B | ≥ | A | + | B | − 1 assuming A , B � ∅ 2 A := { 2 a | a ∈ A } kA = { ka | a ∈ A } Example: 2 Z = CMSC-37115 Mathematical Reasoning

Sumset Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≥ 7 Example when | A + B | = 7 ? A = { 0 , 1 , 2 } , B = { 0 , 1 , 2 , 3 , 4 } , A + B = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } HW (1/12) | A + B | ≥ | A | + | B | − 1 assuming A , B � ∅ 2 A := { 2 a | a ∈ A } kA = { ka | a ∈ A } Example: 2 Z = { even numbers } CMSC-37115 Mathematical Reasoning

Sumset Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≥ 7 Example when | A + B | = 7 ? A = { 0 , 1 , 2 } , B = { 0 , 1 , 2 , 3 , 4 } , A + B = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } HW (1/12) | A + B | ≥ | A | + | B | − 1 assuming A , B � ∅ 2 A := { 2 a | a ∈ A } kA = { ka | a ∈ A } Example: 2 Z = { even numbers } 2 Z + 1 = CMSC-37115 Mathematical Reasoning

Sumset Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≥ 7 Example when | A + B | = 7 ? A = { 0 , 1 , 2 } , B = { 0 , 1 , 2 , 3 , 4 } , A + B = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } HW (1/12) | A + B | ≥ | A | + | B | − 1 assuming A , B � ∅ 2 A := { 2 a | a ∈ A } kA = { ka | a ∈ A } Example: 2 Z = { even numbers } 2 Z + 1 = { odd numbers } CMSC-37115 Mathematical Reasoning

Sumset Question. | A | = 3 , | B | = 5 = ⇒ | A + B | ≥ 7 Example when | A + B | = 7 ? A = { 0 , 1 , 2 } , B = { 0 , 1 , 2 , 3 , 4 } , A + B = { 0 , 1 , 2 , 3 , 4 , 5 , 6 } HW (1/12) | A + B | ≥ | A | + | B | − 1 assuming A , B � ∅ 2 A := { 2 a | a ∈ A } kA = { ka | a ∈ A } Example: 2 Z = { even numbers } 2 Z + 1 = { odd numbers } DO Find a small set B s.t. 2 Z + B = Z CMSC-37115 Mathematical Reasoning