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A set B is a collection of objects such that for every object X in the universe the statement: “ X is a member of B ” Is a proposition. SETS. A quick review of basic notation and set operations. A = {1, 2, ab, ba, 3, moshe, table},

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slide1
A set B is a collection of objects such that for every object X in the universe the statement:

“X is a member of B”

Is a proposition.

SETS
a quick review of basic notation and set operations
A quick review of basic notation and set operations.
  • A = {1, 2, ab, ba, 3, moshe, table},
  • 1,2, ab, ba, moshe table are “elements.” They are members of the set A or “belong” to A.

Notation: ab  A a  A

3. V = {a, i, o, u, e} Set of Vowels

O = {1,3,5,7,9} Odd numbers < 10.

4. A1 = {2, 5, 8, 11, …, 101}

A2 = {1, 2, 3, 5, 8, 13,…}

A3 = {2, 5, 10, 17, 26, …, 101}

basic notation
Basic notation.
  • Set Builder: B = {x | P(x)}

B1 = {x | x = n2 + 1, 1  n  10} (B1 = A3)

B2 = {p | p prime, p = n! + 1, n 

  • Special sets:

N (non-negative integers, natural numbers)

    • Q (rational numbers)
    • Z (integers)
    • Z+ (positive integers)
    • R (real numbers)
    •  (the empty set)
relations among sets
Relations among sets
  • A = B: if x  A  x  B
  • Subsets: A B A  B A  B A  B
  • For every set B:  B
  • A set may have other sets as members:

A = {, {a}, {b}, {a,b}}. Note: A has 4 elements.  A and also  A, {{a}}  A, a  A, {a}  A.

set operations
Set operations

Union: A  B = {x | x  A  x  B} (logic “or”)

Intersection: A  B = {x | x  A  x  B} ( “and”)

Set difference: A \ B = {x | x  A  x  B}

Complement of A: A = {a | a  A} or if U is the “universe” then A = U \ A (“not”).

Example: If U = {a,b,c,…,z} and A = {i,o,e,u,a} then A = {n | n is not a vowel}.

Symmetric difference: A  B = (A \ B)  (B \ A) (“xor”)

slide6
The characteristic vector of a set (representing sets in memory):

Let U = {1,2,…, 15}. Let A = { 3,5,11,13} the characteristic vector of A is the binary string 00101 00000 10100.

The characteristic vector 10010 01101 10001 represents the set {1,4,7,8,10,11,15}.

00000 00000 00000 represents .

Note: with this representation the union of two sets is the OR bit operation and the intersection is the AND.

a simple application
A simple application.

Problem: find the smallest integer n that satisfies the following 3 conditions simultaneously:

(n mod 7 = 5), (n mod 11 = 7), (n mod 17 = 9)

Knowing the language “Math” can help us look for information and use various systems to solve this problem. The following exolains how to use SAGE's set operations to solve problems.

slide8

We can create three sets:

1. A = {k | k = 7n + 5, k < 4000}

2. B = {k | k = 11n + 7, k < 4000}

3. C = {k | k = 17n + 9, k < 4000}

We can then ask SAGE to find the intersection of the three sets. The smallest integer in the intersection (provided there is one) will be our solution.

Answer: {502, 1811, 3120, ...}

venn diagrams
Venn Diagrams

Venn Diagrams : a useful tool for representing information. For instance, the various sets that can be formed by the basic set operations can be viewed by a Venn Diagram.

slide10

A

B

C

slide11
Proving set equalities:

Either: x  A  x  B or A  B  B  A.

Example: De Morgan’s law: A  B = A  B

Proof: Let xA  B.

Then: x  A  B.

Or: x  A and x  B

Or: x A and x B

Or: x AB

Conversely, start from the bottom and go up.

QED

Assume M = {1,2,5,9} then

= A1 A2 A5 A9

slide12
Notation:

A1 A2 …  An = {x | x  Ai i = 1, 2, … , n}.

A1 A2 …  An = {x |  ( i, 1i  n) x  Ai.

Use formula to insert intersection.

Assume M = {1,2,5,9} then

= A1 A2 A5 A9

the power set
The Power Set

Definition: The Power set of the set A is:

P(A) = {B | B  A}.

has 0 elements. P() has one element: P() = {}

A = {a} P(A) = {, {a}} P({}) = {, {}}

the cartesian product
The cartesian product

Cartesian product :

A x B = {(a,b) | a  A  b  B}

Can be defined using sets only:

A x B = {{a}, {a,b}| a  A  b  B}

Note: (a,b)  (b,a) if a  b.

Cartesian product of n sets: A1x A2 x … x An = {(a1, a2,…, an) | ai Ai, i = 1,…,n}

relations
Relations

Definition 1: A relation R, (binary relation) between two sets A and B is a subset of

A x B (mathematically speaking: R  A x B).

Definition 2: A relation R on a set A is a subset of A x A.

relations1
Relations

There are two common ways to describe relations on a set or between two sets:

List all pairs belonging to the relation.

Use set builders to describe the pairs.

Example 1: R0 = {(4,3), (9,2), (3,6), (7,5)} is a relation on N. It is also a relation on A x B where A = {4,9,3,7} and B ={3,2,6,5}

more examples
More examples

Example 2: R2 = {(n,k) | n  N and n + k is a prime number}.

Example 3: R3 = {(n,k) | n,k  N and |n – k| is a multiple of 19}.

Example 4: R3 = {(w,m) | w is a woman, m is a man, w dates m}

classification of relations
Classification of relations

These definitions apply to relations on A.

Definition 3: A relation R on A is reflexive

if (a,a)  R a  A.

Definition 4: A relation R on A is symmetric

if (a,b)  R then (b,a)  R.

R is antisymmetric

if(a,b)  R and (b,a)  R only if a = b.

Definition 5: A relation R on a set A is transitive

if (a,b)  R  (b,c)  R then (a,c)  R.

the transitive closure
The transitive closure

Observation: If R1 and R2 are transitive relations on a set A then so is R1 R2.

Proof: Obvious.

Definition 6: The transitive closure of a relation R on a set A is the “smallest” transitive relation R* on A such that R*  R.