Sets & Set Operations
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Transcript Sets & Set Operations
Discussion #21
Sets & Set Operations;
Tuples & Relations
Discussion #21
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Topics
Sets and Set Operations
Definitions
Operations
Set
Laws
Derivations, Equivalences, Proofs
Tuples and Relations
pairs & n-tuples
Cartesian Product
Relations subset of the cross product
Tuples
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Sets
Sets are collections
The things in a collection are called elements or members
Sets have no duplicates.
Notation { }
Special sets
N natural numbers {0, 1, 2, …} (some exclude 0 from this set)
Z integers; R reals
“set builder” 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
{ x | P(x)} all elements in U that satisfy predicate P
{ x | x>5 x<10} = {6, 7, 8, 9} when U = N
Element of: x A
Cardinality
|A| or #A
both denote the number of elements in A, e.g. |{a,b}| = 2
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Set Equality, Subsets, Supersets
Set Equality
A = B if A and B have the same elements
A = B xA xB
Subsets
B xA xB (subset or equal)
A B A B x(xB xA) (proper subset)
A
Supersets
A B if B A
A B if B A
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Proofs about Set Equality
and the Empty Set
Prove: A = B iff A B B A
A=B
xA xB
definition of set equality
(xA xB) (xB xA) P Q (P Q) (Q P)
ABBA
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
T
Discussion #21
in the empty set, contrary to the defintion)
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Set Operations: Intersection
Intersection
A B {x | xA xB}
{1, 2, 3} {2, 3, 4} = {2, 3}
A
B
Prove: A B A
By definition, A B A xAB xA
1. xA
2. xAB
3. xA xB
4. xA
5. xA xA
6. F
Discussion #21
assume negation of conclusion
premise
def of
3, simplification
1&4, conjunction
5, contradiction
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Set Operations: Intersection
Intersection
A B {x | xA xB}
{1, 2, 3} {2, 3, 4} = {2, 3}
A
B
Prove: A B A
By definition, A B A xAB xA
1. xA
2. xAB
3. xA xB
4. xA
5. xA xA
6. F
Discussion #21
assume negation of conclusion
premise
A simpler proof.
def of
3, simplification
1&4, conjunction
5, contradiction
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Set Operations: Union
Union
A B {x | xA xB}
{1, 2, 3} {2, 3, 4} = {1, 2, 3, 4}
No duplicates!
A
B
Prove: A A B
By definition, A AB xA xA xB
1. xA
2. xA xB
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premise
1, law of addition
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Set Operations: Set Difference
Difference (minus)
A – B {x | xA xB}
{1, 2, 3} – {2, 3, 4} = {1}
Remove elements of B from A
A
B
Prove: A – B A
By definition, A – B A xA–B xA
1. x A – B
premise
2. x A x B
definition
3. x A
simplification
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Set Operations: Complement
Complement
~ A U – A {x | xU xA}
~{1, 2, 3} = {4} if U = {1, 2, 3, 4}
A
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 ~
Fx
comm., contradict., dominat.
T
Note: Unary operators have precedence over binary operators. Use
parentheses for the rest. Possible to define precedence: ~, , , .
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Basic Set Identities
Set Algebra
Name
A ~A = U
A ~A =
AU=A
A=A
Complementation law
Exclusion law
Identity laws
AU=U
A=
Domination laws
AA=A
AA=A
Idempotent laws
Duals: and E
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Basic Set Identities (continued…)
Set Algebra
Name
~(~A) = A
Double
Complement
Commutative laws
AB=BA
AB=BA
(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)
~ (A B) = ~A ~B
~ (A B) = ~A ~B
Distributive laws
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De Morgan’s laws
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Example: Set Laws
Absorption
A (A B) = A
A (A B) = A
A
B
Venn Diagram “Proof”
Prove: A (A B) = A
A (A B)
= (A ) (A B)
= A ( B)
=A
=A
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ident.
distrib.
dominat.
ident.
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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
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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
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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
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Relations
Relation
Subset
of the cross product
Not necessarily a proper subset
R 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
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