Discrete Mathematics with Applications Chapters 6-8: Sets, - PowerPoint PPT Presentation

Set Properties and Boolean Algebras Relations and Functions Discrete Mathematics with Applications Chapters 6-8: Sets, Relations, and Functions (part 2) March 25, 2019 Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions Some Important Subset Relations Inclusion of Intersection: For all sets A and B , A ∩ B ⊆ A and A ∩ B ⊆ B . Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions Some Important Subset Relations Inclusion of Intersection: For all sets A and B , A ∩ B ⊆ A and A ∩ B ⊆ B . Inclusion in Union: For all sets A and B , A ⊆ A ∪ B and B ⊆ A ∪ B . Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions Some Important Subset Relations Inclusion of Intersection: For all sets A and B , A ∩ B ⊆ A and A ∩ B ⊆ B . Inclusion in Union: For all sets A and B , A ⊆ A ∪ B and B ⊆ A ∪ B . Transitivity of Set Inclusion: For all sets A , B , and C , if A ⊆ B and B ⊆ C , then A ⊆ C . Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions Some Important Subset Relations Inclusion of Intersection: For all sets A and B , A ∩ B ⊆ A and A ∩ B ⊆ B . Inclusion in Union: For all sets A and B , A ⊆ A ∪ B and B ⊆ A ∪ B . Transitivity of Set Inclusion: For all sets A , B , and C , if A ⊆ B and B ⊆ C , then A ⊆ C . Next up is a big list of set identities. Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions Hmmm, Looks Familiar... Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions Side by Side Comparison Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions The algebraic structure of the set of statement forms with the logical connectives ∨ , ∧ , and ∼ and the algebraic structure of the set of subsets of a universal set with operations ∪ , ∩ , and c appear to be, in a certain sense, identical. Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions The algebraic structure of the set of statement forms with the logical connectives ∨ , ∧ , and ∼ and the algebraic structure of the set of subsets of a universal set with operations ∪ , ∩ , and c appear to be, in a certain sense, identical. Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions The algebraic structure of the set of statement forms with the logical connectives ∨ , ∧ , and ∼ and the algebraic structure of the set of subsets of a universal set with operations ∪ , ∩ , and c appear to be, in a certain sense, identical. 1 “ ∨ ” (or) corresponds with “ ∪ ” (union) Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions The algebraic structure of the set of statement forms with the logical connectives ∨ , ∧ , and ∼ and the algebraic structure of the set of subsets of a universal set with operations ∪ , ∩ , and c appear to be, in a certain sense, identical. 1 “ ∨ ” (or) corresponds with “ ∪ ” (union) 2 “ ∧ ” (and) corresponds with “ ∩ ” (intersection) Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions The algebraic structure of the set of statement forms with the logical connectives ∨ , ∧ , and ∼ and the algebraic structure of the set of subsets of a universal set with operations ∪ , ∩ , and c appear to be, in a certain sense, identical. 1 “ ∨ ” (or) corresponds with “ ∪ ” (union) 2 “ ∧ ” (and) corresponds with “ ∩ ” (intersection) 3 “ ∼ ” (not) corresponds with “ c ” (complement) Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions The algebraic structure of the set of statement forms with the logical connectives ∨ , ∧ , and ∼ and the algebraic structure of the set of subsets of a universal set with operations ∪ , ∩ , and c appear to be, in a certain sense, identical. 1 “ ∨ ” (or) corresponds with “ ∪ ” (union) 2 “ ∧ ” (and) corresponds with “ ∩ ” (intersection) 3 “ ∼ ” (not) corresponds with “ c ” (complement) 4 “ t ” (tautology) corresponds with “ U ” (universal set) Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions The algebraic structure of the set of statement forms with the logical connectives ∨ , ∧ , and ∼ and the algebraic structure of the set of subsets of a universal set with operations ∪ , ∩ , and c appear to be, in a certain sense, identical. 1 “ ∨ ” (or) corresponds with “ ∪ ” (union) 2 “ ∧ ” (and) corresponds with “ ∩ ” (intersection) 3 “ ∼ ” (not) corresponds with “ c ” (complement) 4 “ t ” (tautology) corresponds with “ U ” (universal set) 5 “ c ” (contradiction) corresponds with “ ∅ ” (empty set) Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions The algebraic structure of the set of statement forms with the logical connectives ∨ , ∧ , and ∼ and the algebraic structure of the set of subsets of a universal set with operations ∪ , ∩ , and c appear to be, in a certain sense, identical. 1 “ ∨ ” (or) corresponds with “ ∪ ” (union) 2 “ ∧ ” (and) corresponds with “ ∩ ” (intersection) 3 “ ∼ ” (not) corresponds with “ c ” (complement) 4 “ t ” (tautology) corresponds with “ U ” (universal set) 5 “ c ” (contradiction) corresponds with “ ∅ ” (empty set) This is not a coincidence! Both are special cases of the same general structure, known as a Boolean algebra. Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions The proofs of these Boolean algebra properties are left to you as an exercise. Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions The proofs of these Boolean algebra properties are left to you as an exercise. Do observe that each of the paired statements can be obtained from the other by interchanging all of the +’s and · ’s and interchanging 1 and 0. Such interchanges transform any Boolean identity into its dual identity. Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions The proofs of these Boolean algebra properties are left to you as an exercise. Do observe that each of the paired statements can be obtained from the other by interchanging all of the +’s and · ’s and interchanging 1 and 0. Such interchanges transform any Boolean identity into its dual identity. It can be proved that the dual of any Boolean identity is also an identity. This fact is often called the duality principle for a Boolean algebra. Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions Example Theorem For all sets A , B , and C , ( A ∪ B ) − C = ( A − C ) ∪ ( B − C ) . Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications

Set Properties and Boolean Algebras Relations and Functions Example Theorem For all sets A , B , and C , ( A ∪ B ) − C = ( A − C ) ∪ ( B − C ) . Prove this theorem 1 by showing they’re subsets of each other. 2 using Boolean algebra identities. Chapters 6-8: Sets, Relations, and Functions (part 2) Discrete Mathematics with Applications