1Discussion #21

Discussion #21

Sets & Set Operations;Tuples & Relations

2Discussion #21

Topics

Sets and Set OperationsDefinitionsOperationsSet LawsDerivations, Equivalences, Proofs

Tuples and RelationsTuples pairs & n-tuplesCartesian ProductRelations subset of the cross product

3Discussion #21

Sets Sets are collections

The things in a collection are called elements or members Sets have no duplicates.

Notation { } Enumerate: {1, 2, 3} Ellipsis: {1, 2, …} or {1, 2, … , 100} Universe: U, universe of discorse Empty set: { } or i.e. set with no elements

Special sets NN natural numbers {0, 1, 2, …} (some exclude 0 from this set) ZZ integers; RR reals

“set builder” notation { x | P(x)} all elements in U that satisfy predicate P { x | x>5 x<10} = {6, 7, 8, 9} when U = NN

Element of: x A Cardinality

|A| or #A both denote the number of elements in A, e.g. |{a,b}| = 2

4Discussion #21

Set Equality, Subsets, Supersets Set Equality

A = B if A and B have the same elementsA = B xA xB

SubsetsA B xA xB (subset or equal)A B A B x(xB xA) (proper subset)

SupersetsA B if B A

A B if B A

5Discussion #21

Proofs about Set Equalityand the Empty Set

Prove: A = B iff A B B AA = B xA xB definition of set equality

(xA xB) (xB xA) P Q (P Q) (Q P) A B B A definition of subset

Prove: A (i.e. is a subset of every set.) A x xA definition of subset

F xA x is false (for if not there is an element of U in the empty set, contrary to the defintion)

T

6Discussion #21

Set Operations: IntersectionIntersection

A B {x | xA xB}{1, 2, 3} {2, 3, 4} = {2, 3}

Prove: A B ABy definition, A B A xAB xA

1. xA assume negation of conclusion

2. xAB premise3. xA xB def of 4. xA 3, simplification5. xA xA 1&4, conjunction6. F 5, contradiction

A B

7Discussion #21

Set Operations: IntersectionIntersection

A B {x | xA xB}{1, 2, 3} {2, 3, 4} = {2, 3}

Prove: A B ABy definition, A B A xAB xA

1. xA assume negation of conclusion

2. xAB premise3. xA xB def of 4. xA 3, simplification5. xA xA 1&4, conjunction6. F 5, contradiction

A B

A simpler proof.

8Discussion #21

Set Operations: Union

UnionA B {x | xA xB}{1, 2, 3} {2, 3, 4} = {1, 2, 3, 4}No duplicates!

Prove: A A BBy definition, A AB xA xA xB

1. xA premise2. xA xB 1, law of addition

A B

9Discussion #21

Set Operations: Set Difference

A B

Difference (minus)A – B {x | xA xB}{1, 2, 3} – {2, 3, 4} = {1}Remove elements of B from A

Prove: A – B ABy definition, A – B A xA–B xA

1. x A – B premise

2. x A x B definition

3. x A simplification

10Discussion #21

Set Operations: Complement

Complement~ A U – A {x | xU xA}~{1, 2, 3} = {4} if U = {1, 2, 3, 4}

Prove: A ~A = A ~A =

A ~A A ~A set equality A ~A T is a subset of every set A ~A identity x A x ~A x def of and x A x U x A x def of ~ F x comm., contradict., dominat. T

Note: Unary operators have precedence over binary operators.Use parentheses for the rest. Possible to define precedence: ~, , , .

A

11Discussion #21

Basic Set IdentitiesSet Algebra Name

A ~A = UA ~A =

Complementation lawExclusion law

A U = AA = A

Identity laws

A U = UA =

Domination laws

A A = AA A = A

Idempotent laws

Duals: and E

12Discussion #21

Basic Set Identities (continued…)Set Algebra Name

~(~A) = A Double Complement

A B = B A A B = B A

Commutative laws

(A B) C = A (B C) (A B) C = A (B C)

Associative laws

A (B C) = (A B) (A C) A (B C) = (A B) (A C)

Distributive laws

~ (A B) = ~A ~B~ (A B) = ~A ~B

De Morgan’s laws

13Discussion #21

Example: Set Laws Absorption

A (A B) = AA (A B) = A

Venn Diagram “Proof”

Prove: A (A B) = A A (A B)

= (A ) (A B) ident.= A ( B) distrib.= A dominat.= A ident.

A B

14Discussion #21

Tuples Things (usually a small number of things) arranged

in order 2-tuples

pairs (x, y) ordered (x, y) (y, x) unless x = y

n-tuples = (x1, x2, …, xn) Typically, elements in tuples are taken from known

sets x females, y males

(Mary, Jim) e.g. might mean: Mary and Jim are a married couple x people, y cars

(Mary, red sports car17) e.g. might mean: Mary owns red sports car17

x, y, z integers(3, 4, 7) e.g. might mean: 3 + 4 = 7

15Discussion #21

Cartesian Product

A B = {(x, y) | xA yB}

e.g. A = {1, 2}B = {a, b, c}A B = {(1, a), (1, b), (1, c),

(2, a), (2, b), (2, c)} |A B| = |A| · |B| = 2 · 3 = 6

16Discussion #21

Cartesian Product (continued…) n-fold Cartesian Product

A1 … An = {(x1, …, xn) | xA1 … xnAn}e.g. A = {1, 2}

B = {a, b, c}C = {, }

A B C = {(1,a,), (1,a,), (1,b,), (1,b,), (1,c,), (1,c,), (2,a,), (2,a,), (2,b,), (2,b,), (2,c,), (2,c,)}

Can get large:A = set of students at BYU (30,000)B = set of BYU student addresses (10,000)C = set of BYU student phone#’s (60,000)|A| |B| |C| = 1.8 1013

17Discussion #21

Relations Relation

Subset of the cross productNot necessarily a proper subsetR A B or R A B C

Examples:A = {1, 2} & B = {a, b, c}

R = {(1, a), (2, b), (2, c)}A = {1, 2} & B = {a, b, c} & C = {, }

R = {(1, a, ), (2, c, )}Marriage: subset of the cross product of

males and females