Sets. Ordered pairs.

Sets

Ordered pair

p. 2

Sets

The set theory is a branch of mathematical logic thatwas created by Georg Cantor in 1870s.

Def. A set is a unordered collection of objects beingregarded as a single object.

Examples:A= {1, 2,3, 4,5}

B = {x ∈ A | (x ≥ 3)∧ (x is odd)}

C = {∅, {A}, {B}, {A, B}}

N= {0, 1,2, 3, . . .}

Sets

Ordered pair

p. 3

Sets

If x belongs to a set A, we say that it is a member (or an element)of A and write

x ∈ A.

If x is not a member of A, we write

x /∈ A.

Empty set, denoted by ∅ or ;, has no members:

∀x (x /∈∅).

Sets

Ordered pair

p. 4

Sets

Two sets A and B are equal, A = B, iff they have exactly the sameelements:

∀x (x ∈ A↔ x ∈ B)

For any two objects x and y , we can make a set containing exactlythese two objects

{x , y}

If those two objects are identical, x = y , we get a singleton set,

{x , x}= {x}.

Notice that these sets are not equal:

∅, {∅}, {∅, {∅}}

Sets

Ordered pair

p. 5

Sets

Set-builder notation.

A set of all objects that satisfy the property P:

A= {x | P(x)}

Examples:B = {x ∈ Z | x is even}

C = {x ∈ Z | ∃k ∈ Z (x = 2k)}

D = {x | (x ∈ Z)∧ (∃k ∈ Z (x = 2k))}

E = {0,2,−2,4,−4, 6,−6, . . .}

In naive set theory, any definable collection is a valid set. And usu-ally it works fine.

However, it leads to contradictions, such as Russell’s paradox.

Sets

Ordered pair

p. 6

Russell’s paradox

Let R be the set of all sets that are not members of themselves:

R= {x | x /∈ x}

It is legal to ask, is R a member of itself or not.

There are only two cases, either R ∈ R, or R /∈ R.

So, where is the paradox?

Sets

Ordered pair

p. 7

Russell’s paradox

R= {x | x /∈ x}

(a) If R is a member of itself, then by its definition, R /∈ R,

R ∈ R→ R /∈ R.

(b) Otherwise, if R is not a member of itself, then R ∈ R,

R /∈ R→ R ∈ R.

Therefore, we get a contradiction, R ∈ R↔ R /∈ R.

There exist several axiomatic systems that rule out such pathologi-cal cases. For example, Zermelo-Fraenkel set theory with the Axiomof Choice (ZFC).

Sets

Ordered pair

p. 8

Union and Intersection

Union of two sets A and B,

A∪ B = {x | (x ∈ A)∨ (x ∈ B)}

{1,2} ∪ {2, 3}= {1,2, 3}

Intersection of two sets A and B,

A∩ B = {x | (x ∈ A)∧ (x ∈ B)}

{1, 2} ∩ {2, 3}= {2}

Sets

Ordered pair

p. 9

Difference

Difference between two sets A and B,

A\ B = {x | (x ∈ A)∧ (x /∈ B)}

A B

{1, 2} \ {2, 3}= {1}

Sets

Ordered pair

p. 10

Complement

If there is a universal set U of all possible objects, then the comple-ment of a set A is

A= U \ A= {x ∈ U | x /∈ A}= {x ∈ U | ¬(x ∈ A)}

AA

U

Exmaple: When U = Z:

Odd = {x ∈ Z | x is odd}

Even= Odd = Z \Odd

Sets

Ordered pair

p. 11

Set identities

Given the universal set U , sets with respect to union, intersection,and complement satisfy the same identities as propositions withrespect to ∧, ∨, and ¬

Sets: A∩ B A∪ B A U ∅Propositions: p ∧ q p ∨ q ¬p T F

A = A

A∪ B = A∩ B

A∩ B = A∪ BA∩ (B ∩ C) = (A∩ B)∩ CA∪ (B ∪ C) = (A∪ B)∪ CA∩ (B ∪ C) = (A∩ B)∪ (A∩ C)A∪ (B ∩ C) = (A∪ B)∩ (A∪ C)

See the full list in Rosen’s book.

Sets

Ordered pair

p. 12

Set identities

Let’s prove De Morgan’s law for sets:

A∪ B = A∩ B

(x ∈ A∪ B) = ¬(x ∈ A∪ B)= ¬((x ∈ A)∨ (x ∈ B))

(x ∈ A∩ B) = (x ∈ A)∧ (x ∈ B)= ¬(x ∈ A)∧¬(x ∈ B)= ¬((x ∈ A)∨ (x ∈ B))

The propositions in the right hand sides of the equations are equal,therefore, the left hand sides are equal too.

Sets

Ordered pair

p. 13

Subset

A is a subset of B iff every element of A is an element of B:

A⊆ B ↔�

∀x((x ∈ A)→ (x ∈ B))�

{1,2} ⊆ {1,2, 3}

{1, 2,3} ⊆ {1, 2,3}

∅ ⊆ {1, 2,3}

A is a proper subset of B iff A is a subset of B, but it’s not equal to B

A( B ↔�

∀x((x ∈ A)→ (x ∈ B)) ∧ ∃x((x ∈ B)∧ (x /∈ A))�

{1,2} ( {1,2, 3}

∅ ( {1, 2,3}

Proper subset A is strictly “smaller” than B.

Sets

Ordered pair

p. 14

Power set

Def. The set of all subsets of A is called a power set of A, denotedby P (A).

Examples:

P ({0, 1}) =�

∅, {0}, {1}, {0, 1}

P ({0, 1, 2}) =�

∅, {0}, {1}, {2}, {0,1}, {0, 2}, {1, 2}, {0,1, 2}

Sets

Ordered pair

p. 15

Power set

Def. The set of all subsets of A is called a power set of A, denotedby P (A).

Examples:

P ({0, 1}) =�

∅, {0}, {1}, {0, 1}

P ({0, 1, 2}) =�

∅, {0}, {1}, {2}, {0,1}, {0, 2}, {1, 2}, {0,1, 2}

Sets

Ordered pair

p. 16

Cardinality

Def. The cardinality of a finite set A is equal to the number of ele-ments in A. It’s denoted by |A|.

|∅|= 0

|{0,1, 2,3, 4}|= 5

We already know the subtraction rule for the cardinality of a union:

|A∪ B|= |A|+ |B| − |A∩ B|

Sets

Ordered pair

p. 17

Question

Compute the cardinality of the power set P (A) if |A|= n.

In other words, since the power set is the set of all subsets, the task isto count the number of subsets of a set with n elements.

|P (A)|= 2|A| = 2n

Sets

Ordered pair

p. 18

Question

Compute the cardinality of the power set P (A) if |A|= n.

In other words, since the power set is the set of all subsets, the taskis to count the number of subsets of a set with n elements.

|P (A)|= 2|A| = 2n

Sets

Ordered pair

p. 19

Question

Compute the cardinality of the power set P (A) if |A|= n.

In other words, since the power set is the set of all subsets, the taskis to count the number of subsets of a set with n elements.

|P (A)|= 2|A| = 2n

Sets

Ordered pair

p. 20

Generalized union and intersection

Union of a finite set of sets:⋃

{A0}= A0

⋃

{A0, A1, . . . An}=n⋃

i=0

Ai = A0 ∪ A1 ∪ . . .∪ An

Intersection of a finite set of sets:⋂

{A0}= A0

⋂

{A0, A1, . . . An}=n⋂

i=0

Ai = A0 ∩ A1 ∩ . . .∩ An

Sets

Ordered pair

p. 21

Ordered pair

Def. The ordered pair of a ∈ A and b ∈ B is an ordered collection(a, b).

Two ordered pairs are equal

(a, b) = (c, d) if and only if (a = c)∧ (b = d).

Observe that this property implies that

(a, b) 6= (b, a)

unless a = b. So, the order matters. This is why it is called theordered pair, and (a, b) is not equivalent to a set {a, b}.

(1, 2) = (1, 2)(1, 2) 6= (1, 3)(1, 2) 6= (2, 1)

Sets

Ordered pair

p. 22

Ordered pair

More examples of ordered pairs:

(1, 2)({1}, {2})(1, {2, 3,4, 5})

((1, 2),∅)

If a ∈ A and b ∈ B, what is the set of all ordered pairs (a, b)?

Sets

Ordered pair

p. 23

Cartesian product

Def. Let A and B be sets. The Cartesian product of A and B, denotedby A×B, is the set of all ordered pairs (a, b), where a ∈ A and b ∈ B.Hence,

A× B = {(a, b) | a ∈ A∧ b ∈ B}.

(Named after Rene Decartes)

Question. Given two sets A= {1, 2,3} and B = {C , D}, what is theirCartesian product A× B?

A× B = {(1, C), (2, C), (3, C),(1, D), (2, D), (3, D)}

Sets

Ordered pair

p. 24

Cartesian product

Def. Let A and B be sets. The Cartesian product of A and B, denotedby A×B, is the set of all ordered pairs (a, b), where a ∈ A and b ∈ B.Hence,

A× B = {(a, b) | a ∈ A∧ b ∈ B}.

(Named after Rene Decartes)

Question. Given two sets A= {1, 2,3} and B = {C , D}, what is theirCartesian product A× B?

A× B = {(1, C), (2, C), (3, C),(1, D), (2, D), (3, D)}

Sets

Ordered pair

p. 25

Cartesian product

Question. If the |A| = n, and |B| = m, what is the cardinality ofA× B?

Sets

Ordered pair

p. 26

Building a list

Ordered pair is fundamental for defining data types.Question. How to implement the list data type using only orderedpairs?

Example of a list:[1, 2,3, 4]

Interface:construct ( 1, [2, 3,4] ) = [1, 2,3, 4]

head ( [1, 2,3, 4] ) = 1

tail ( [1, 2,3, 4] ) = [2, 3,4]

Possible implementation:(1, (2, (3,4)))

construct ( h, t )= (h, t)head ( (h, t) )= h

tail ( (h, t) )= t

Sets

Ordered pair

p. 27

Building a list

Ordered pair is fundamental for defining data types.Question. How to implement the list data type using only orderedpairs?

Example of a list:[1, 2,3, 4]

Interface:construct ( 1, [2, 3,4] ) = [1, 2,3, 4]

head ( [1, 2,3, 4] ) = 1

tail ( [1, 2,3, 4] ) = [2, 3,4]

Possible implementation:(1, (2, (3,4)))

construct ( h, t ) = (h, t)head ( (h, t) ) = h

tail ( (h, t) ) = t

Sets

Ordered pair

p. 28

Ordered n-tuple

Def. The ordered n-tuple of is an ordered collection (a1, a2, . . . , an).

It is just an extension of an ordered pair for joining n elementstogether.

Def. The Cartesian product of the sets A1, . . . An, is the set of alln-tuples such that

A1 × . . .× An = {(a1, . . . , an) | ai ∈ Ai for i = 1, . . . , n}

If all sets Ai are equal, that is, A1 = . . . = An = A, then their Carte-sian product is denoted by An

A1 × . . .× An = A× . . .× A︸ ︷︷ ︸

n times

= An