cs2102: Discrete Mathematics Class 7: Sets, SAT problem David Evans, Mohammad Mahmoody University of Virginia

Plan • Sets (as a Data Type) • Review: SAT problem and its importance Thursday: continue other data types..

Type Operations

Data Types Mathematical • Defines a set of possible values and operations on those values • Finite or infinite Programming Languages • Defines a set of possible values and operations on those values • Finite (in practice), but often “viewed” as infinite

Sets as Data Type Unordered collection of objects Infinite or finite Objects may be of any type not necessarily the same including sets 4 2102 { 1, 2, 3, …} Each object only counts once (multiset has count for each object)

Membership (∈) ∈ is true iff is in the set 4 2102 { 1, 2, 3, …}

Not being a member (∉) ∉ is true iff is not in the set 4 2102 { 1, 2, 3, …}

Subset (⊆) ⊆ iff every element of A is an element of B. 4 2102 { 1, 2, 3, …}

Set Equality How should we define = for sets? 4 2102 { 1, 2, 3, …}

Set Equality = ⟺ ⊆ ∧ ⊆ 4 2102 { 1, 2, 3, …} How should we define = for sets?

More Set Operations Union: ∪ Intersection: ∩ Difference: − A B

Defining Union Union: ∪ A B

Defining Intersection A B Intersection: ∩

Defining Difference A B Difference: −

Complement Complement: ҧ A ∀ ∈ . ∈ ⟺ ∉ ҧ . D = “domain of discourse” (Universe) B U 4 2102 { 1, 2, 3, …}

∈ ⟺ is in the set Membership ∉ ⟺ ¬( ∈ ) ∈ ҧ ⟺ ( ∈ ∧ ∉ ) Complement ∀. [ ∈ − ⟺ ∈ ∧ ∉ ] Difference ∀. [ ∈ ∩ ⟺ ∈ ∧ ∈ ] Intersection ∀. [ ∈ ∪ ⟺ ∈ ∨ ∈ ] Union ⊆ ⟺ ∀ ∈ . ∈ Subset Equality = ⟺ ⊆ ∧ ⊆ Non-membership Which ones are necessary?

B Exchanging Union wtith Intersection? A B U

Self-Membership ? Is there any set where ∈ ? Is it even meaningful to ask ∈ ?

Conundrum notself = the set of all sets that are not members of themselves Is notself ∈ notself ? all = the set of all sets notself is a subset of all .

notself = the set of all sets that are not members of themselves Is notself ∈ notself ? Slack break…

notself = the set of all sets that are not members of themselves Is notself ∈ notself ? Russell’s Paradox! IF Yes, it means notself ∈ notself , so notself is a member of itself and should not be in notself . IF No, it means notself ∉ notself , so notself is not a member of itself and should be in notself .

Where did we make a mistake?

Conundrum notself = the set of all sets that are not members of themselves Is notself ∈ notself ? all = the set of all sets notself is a well-defined subset of all .

Sets as data type Unordered collection of objects Infinite or finite Objects may be of any type... 4 2102 { 1, 2, 3, …} Apple

Charge • PS3 Due this Friday (6:29pm) start early • Next session: Other Data Types – Sequences, Functions, etc. – Read MCS Chapter 4